FORCED RESPONSE COMPUTATION FOR BLADED DISKS INDUSTRIAL PRACTICES AND ADVANCED METHODS

Transcription

1 FORCED RESPONSE COMPUTATION FOR BLADED DISKS INDUSTRIAL PRACTICES AND ADVANCED METHODS E. Seinturier * Turbomeca (SAFRAN) Bordes, France Abstract The aero-mechanical optimisation of turbomachinery bladed disk is directly driving its final efficiency and reliability. The typical design process is presented here and improvements are proposed, based on forced response assessment. It is shown that this prediction is difficult because many parameters are required and all are not necessary known in the earlier steps of the design process. Moreover, different specificity of bladed disks must be modelled to represent accurately their physical behaviour (viscous flows, aeroelastic couplings, mistuning, non linear dynamic behaviour). Nevertheless, these analyses can provide a useful help to drive the designer choices. Keywords: aeromechanical compromise, bladed disks dynamics, forced response, turbomachinery I. Introduction Turbomachinery designers have a constant preoccupation: improve performance while reducing masses. The new aerodynamic computation tools (CFD) are leading to new types of blade geometry. Indeed, 3D and viscosity effects, now taken into account in the CFD design, generates complex and thinner blade shapes. One main consequence, often underestimated, is the effect on the blade stress design and dynamic behaviour. Static loads due to the rotation speed, the temperature and the steady aerodynamic forces are now so important that the ability of the material to accept extra dynamic stresses is dramatically reduced. Moreover, the intensity of the sources of excitation in turbomachines has increased: smaller axial gaps between stators and rotors, smaller tip gaps inducing rotor/casing contact risks, etc... A significant illustration of that is the analysis of failure problems under operating condition: nowadays, the majority of problems encountered is connected to High Cycle Fatigue (HCF). These problems are often very sudden because a huge number of stress cycles can be performed in a very short time (during one flight for instance). A 1 khz vibration during 20 minutes corresponds to one million oscillations. The margins for dynamic loads being smaller and smaller, the tolerance to vibration have become a key point. The engine manufacturer has two main approaches to mitigate HCF risks: - The first approach is the standard design practice, which consists in avoiding dangerous resonance in the operating range. The resonance dangerousness is very empirical and based on older engine experience. In general resonance must be avoided on the first modes (firsts bending and torsion modes). But it is impossible to remove all the resonance and consequently an engine testing is required in order to assess the dynamic levels on the remaining ones. These tests are very expensive and are performed at the very end of the development process. The late discovery of vibratory problems can impact dramatically the engine development schedule and costs. - The second approach is to accept resonance in the operating range but to estimate early in the design process the associated response level. This requires specific methods for forced response computation. Moreover, in order to improve the accuracy of the analysis, many secondary phenomenon must be accounted for, such as mistuning or friction damping. In this case, advanced methods are necessary. II. Mechanical particularities of bladed disks Bladed disks are rotating structures submitted to high load levels (static, dynamic and impact). The way to account for steady load will be presented first. But these structures are also cyclically symmetric. The mechanical couplings between the blade and the disk are important: while few years ago the blade alone was designed, it is now necessary to study the complete assembly. The computational cost of a 3D model of a 80 blades bladed disks is very important (few millions degrees of freedom). The structure symmetry properties are used to reduce the cost of the analysis by modelling one blade and one repetitive disk sector. The techniques used are presented in this chapter. 1

2 A. Rotation effects - Static analysis Only one sector of the structure is meshed. On each cut of the disk, specific symmetry conditions are used. For the static analysis, these conditions are simple: the displacements of the two disk cut faces must be similar in order to guaranty the structure continuity. The problem to solve is a simple static linear problem: Fs = [K].X The unknown is X, the static displacement vector of the structure, [K] is the stiffness matrix and Fs the steady forces applied to the structure (thermal, rotation, steady aerodynamic forces ). The application of the rotation forces generates a deformation of the structure. This deformation modifies the rotation forces field, which is dependant of each point space co-ordinate and its associated mass. Two solutions are possible to deal with this difficulty: - In case of significant displacement, typically for fan stages, a non linear analysis is required. The non linearity is due to the significant deformation of the structure that modifies the stiffness matrix and the force field. In such a case, at each iteration of the computation, the deformation is determined and the associated stiffness matrix and forces are updated. The convergence is obtained when the evolution of deformations is small between two steps. At the end of the process, the deformation is identified as well as the structure updated stiffness matrix. - In case of small displacements, only two iterations are generally performed. The deformation is computed and two additive stiffness terms are identified. A [Ks] term, due to the structure deformation than modifies its stiffness. A [Kr] term due to the force field evolution (spin softening). The stiffness matrix is updated ([K1]=[K]+[Kr]+[Ks]) and the problem is solved again but with the basic force field and without geometry update (which are already accounted for in the updated stiffness matrix). It can be seen that one single static analysis does not give an accurate results. The rotation effects are requiring at least two successive linear analyses (not expensive and generally sufficient) or one non linear analysis (for fan for instance). If the rotation effect is not properly taken into account, the analysis can lead to a wrong operating geometry, not consistent with the one determined by the CFD analysis. B. Dynamic analysis For the dynamic analysis, the object is to solve the following equation: M X! ( t) + K X ( t) = [ ] [ ] 0 where [M] is the mass matrix of the structure and X(t), X"(t) respectively the displacement and acceleration of the structure, function of time. [K] is the updated stiffness matrix, obtained in the previous static analysis. To solve this problem, the space and time domains are separated by the following assumption: X ( t) = Xe i! t where X is dependant of space only and ω is the pulsation. The first equation becomes: 2!" M + K X = ( [ ] [ ]) 0 The extraction of eigenvalues and eigenvectors of the previous expression allows to obtain the natural frequencies ω i of the structure and its modeshapes φ i. The natural frequencies of the structure are dependant on the mass and stiffness matrices of the structure. The effect of the rotation speed and thermal loads is significant on the stiffness matrix: the frequencies can nearly double between rest and full speed on a fan stage for instance. Cyclic symmetry: For the static analysis, the modelling of one sector of the structure only is associated to simple boundary conditions. For the dynamic analysis, the structure symmetry must also be taken but in this case it is a bit more complex. Indeed, the assumption that the two cut faces have the same displacement does not apply anymore. The continuity of the structure must be imposed but allowing a constant phase angle between each sector. The different modes of the whole structure with N blades can be classified in function of the phase angle β between each sector, or what is called the number of nodal diameters n. The phase angle and the number of nodal diameters are connected by the following relation: β=2π n/n. The solution domain is completely defined for 0 β π. - If N is pair, the solution space is defined between n=0 and n=n/2. - If N is odd, the solution space is defined between n=0 and n=(n-1)/2 - if the number of nodal diameters n=0, the phase angle β=0, all the blades are oscillating in phase. - if the number of nodal diameters is n=n/2 (N pair), the phase angle β=π/2, all the blades are oscillating out of phase. 2

3 - For any other number of nodal diameter n, β=2π n/n The boundary conditions applied to the sector of the structure are defined as shown on the following Figure 1. The two sectors cut faces nodes are related by a complex boundary condition imposing the same motion with a given constant phase angle. It is clear than one dynamic computation corresponds to one given phase angle. N/2 computations are required to define completely the solution space. But most of the time, not all the phase angles are necessary and some CPU cost reduction techniques are available. Figure 1 - Cyclic symmetry boundary conditions When n=0 or n=π, the solution is real. Else, the solution is complex: indeed, the term isin(β) is not null in the boundary condition, so the stiffness matrix is complex leading to a complex solution. In such a case, the solution is double: it induces that the mode is rotating on the bladed disk. Basically, it can be assumed that the solution is defined modulus 2π. But for, 0 β π. the frequencies and modeshapes are the same than for π β 2π., only the mode rotation direction is different. Consequently, the space solution can be fully represented (in terms of frequencies and modeshape) for 0 β π.. To illustrate the double solution, the case of a clamped circular rod is proposed. The rod section is symmetric (axial symmetry) and a non finite number of first flexion modes is possible, in any direction of space. Moreover, the mode can rotate as shown on the following Figure 2. The cyclically symmetrical modes can be grouped by family. The family is defined as the group of all the modes for a similar blade modeshape. For instance, for the first flexion mode of the blade, (N+1)/2 (if N is pair, (N-1)/2 if N is odd) similar modes corresponding to the various phase angles are in the same mode family. C. Excitations For one given mode family, only few phase angles will be excited, depending on the space repartition of the excitation. The sources of excitation in the engine are numerous and most of them are harmonic. Effectively, blades excitations are mainly due to non uniformity of the upstream pressure field. When rotating, the blades see a fluctuating pressure field at a frequency connected to the rotation speed. The sources of excitation can be: - The upstream stator (with Es blades) generating an excitation frequencies equal to the rotation frequency (ω) times the number of stator blades plus higher harmonic (Es.ω, 2Es.ω, 3Es.ω, etc ) with decreasing intensity. - The downstream stator, mainly for transonic stages, generating the same kind of frequency as before. - For fan stages, there is no upstream stator but the flow in the air inlet is not uniform (aircraft incidence, lateral wind, presence of the aisle up to the engine, presence of the fuselage on one side only, etc ). This generates excitation frequencies from 1N to 6N. - Turbines can be excited also by the number of injector (Ei) in the combustion chamber at a frequency equal to Ei. ω and upper harmonics - Moreover, low engine order are present in the engine, due to static parts non perfect symmetries, such as not perfectly circular casings, not perfectly constant stator vanes or injector spacing, etc If the bladed disk with N blades is excited by E excitations per round at a frequency equal to E times ω (ω is the rotation speed). If F is even, the number of diameter n excited will be: - if E N/2 (if N is odd E (N-1)/2), n=e - if N/2 E N-1 (if N is odd, (N+1)/2 E N), n=n-e - if N E 3N/2 (if N is odd, N E (3N-1)/2), n=e-n - if 3N/2 E 2N (if N odd, (3N+1)/2 E 2N), n=2n-e etc For instance, in the case of N=10 blades, the number of diameter and phase angle (360.n/N modulus 360 ) are given in the following table I. Figure 2 - Illustration of cyclic symmetry on a simple rod 3

4 E N Phase Comment ( ) Real mode Complex mode Complex mode Complex mode Complex mode Real mode = => n=4 (opposite rotation) = => n=3 (opposite rotation) Same Same Same solution as E= Same solution as E=1 TABLE I. Relation between engine order and nodal diameter The following Figure 3 gives a simple way to estimate the excited number of diameters (or phase angle): x-axis is the number of nodal diameters (n), y-axis is the number of excitation per round (E). In this illustration, the number of blades N is supposed to be even. the case for the first modes). Figure 4 - Typical modeshape rebuilt on 1 third of a bladed disk Figure 5 - Frequency diagram - 64 blades Figure 3 - "Zigzag" diagram Figure 4 shows a typical cyclically symmetric modeshape on a turbine bladed disk. The change of the number of nodal diameters is computed by imposing different boundary conditions on one bladed disk sector. This affects the stiffness matrix and consequently the bladed disks frequencies. In other words, the first flexion mode of the blade is not at the same frequency in function of the excitation order. This is illustrated on the following Figure 5 were 15 modes of the bladed disk has been computed for the different possible phase angles (from 0 to 32) of a 64 blades turbine rotor. When the mode is dominated by the blade, with only a small participation of the disk, it is not much affected by the number of nodal diameters (this is III. Basic techniques for blade vibratory aeromechanical design This first chapter presents the standard techniques used in the aeromechanical design of turbomachinery bladed disks. This general principle applies to fans, compressors and turbine blades even if each blade raw has some specificity (hollow turbine blades, blisks, shrouded blades, etc...). One can remark that this design process is highly iterative and each engine manufacturer has its own convergence criteria. A. Aerodynamic optimisation The first step of the design is obviously an aerodynamic proposal of a blade, meeting the objectives of performance (pressure ratio, flow) and efficiency (reduced 4

5 losses). The CFD is now performed in 3D, with steady Navier- Stokes codes. The computational power allows to mesh precisely the airfoil accounting for technologic effects such as tip gaps, air throats defects, etc... Most of the approaches are now based on full stage (stator + rotor) steady computations in order to optimise the whole stage instead of each grid separately. The accuracy of the CFD will depend on the step of the iterative process. If this is the first aerodynamic airfoil proposed, the analysis will be light. Going deeper in the iterations, the modelling will get more complex in order to provide a better assessment of the airfoil performances, mainly taking into account technologic effects such as tip clearances, air throat non continuity, rotor ageing, etc The CFD analysis provides an optimised airfoil "operating" geometry for one or few given nominal operating point (take off for short range aircraft, cruise for long range, etc ). An example of steady flow (iso-mach at 90% span - Navier Stokes 3D analysis) is given on Figure 6 computed. - This deviation is added on the previous cold mesh geometry. - Deformation are computed again, etc The process ends when the cold mesh deformation is sufficiently close to the CFD one. In some case, when the displacements between rest and full speed are small, it has been seen previously that a simple linear analysis can be performed to estimate the airfoil deformations. In such a case, the inverse operation is fairly simple and the cold geometry can be obtained in few iterations. It is much more complex to obtain this cold geometry in case of more important displacements between rest and full speed, for instance for fans, because a non linear analysis is required to compute deformations. At the end of this step, a cold geometry of the airfoil is defined. For the first aeromechanic iteration, the airfoil only is modelled but rapidly a shank, an attach and a disk geometry are proposed in order to account for the disk flexibility in the passage between "cold" and "operating" conditions. Figure 7 shows the example of a passage from "cold" to "operating" conditions on a fan blade. On the left, the blade static deformation is shown. On that type of fan blade, displacements can be significant (few mm). On the right is shown the comparison of the blade tip displacements under operating condition with a linear or non linear analysis. Figure 6 - Example of a steady flow field after CFD optimisation B. Static stresses assessment Cold geometry Once the airfoil geometry is defined, the first objective of the stress designer is to find the "cold" geometry of the airfoil. The objective is to find the geometry at rest (the one that could be manufactured), whose deformation under steady loads is close to the geometry defined by the CFD. An iterative process is used to determine the cold geometry: - steady loads are applied to a FE model based on the CFD geometry. - the deformations are computed - the deformation field is removed from the basic CFD geometry to create a new mesh, corresponding to the cold geometry - the deformations are computed under steady loads. - the deformed geometry is compared to the CFD one and the deviations between the two geometry is Figure 7 - Blade deformation under operating condition Static stresses analysis The cold geometry being known as well as the deformation under steady loads, the stresses can be analysed. Some modifications can be proposed in order to reduce local stresses (for instance at the junction between the blade and the platform). The modification should be minimised in order to reduce the impact on the 5

6 aerodynamic performance of the airfoils. Nevertheless, in some cases, major modifications are required because of the blade shape or thickness, not suitable regarding the stress analysis. Each engine manufacturer has its own criteria for static stress assessment in the aero-mechanical design process. The basic idea is of course to minimise the static stresses for various reasons: - The lower the static stresses are, the higher the allowable dynamic stresses will be. - The blade will have to be certified regarding ingestion and low steady stresses increases the tolerance to Foreign Object Damages (FOD). - At the end of the design process, specific life predictions associated to complex and expensive computations are performed. Lower steady stresses will increase the blade life. Moreover, extra design criteria should be respected. For instance, the blade, under normal operating conditions, should not be solicited in flexion. The skins stresses should be equal on the pressure and suction sides. Finally, the static analysis allows determining blade tip gaps. These clearances must also be minimised in order to increase the bladed disk performances. The static stresses have been assessed. Sometimes, the modification on the airfoil is so important that an aerodynamic re-design of the blade is necessary. Else, the process continues with the dynamic analysis. A typical steady stress (equivalent Von Mises) distribution is shown on the following Figure 8. steady loads increase can have a stiffening effect (rotation speed), or a softening effect (thermal). Consequently, the bladed disk frequencies are changing with the operating point: - For fan and low pressure compressor, the rotation effect is predominant, so the frequency increases with the rotation speed. - For high pressure turbines, when the rotation speed increases, the temperature increases also and the thermal effect becomes predominant. The frequencies are decreasing with speed. - For high pressure compressors or low pressure turbines, the thermal and rotation effects are more or less compensating. The frequency change is small on the operating range. The modal analysis has to be performed for various operating points. The first objective of this analysis is to check that the natural frequencies of the blades can not be excited within the operating range. The tool commonly used by the designer is the Campbell diagram. The x-axis is the rotation speed, the y-axis is the frequency. The bladed disk natural frequencies are plotted in function of speed as well as the straight lines corresponding to the excitation frequencies. The crossing between the excitation and the natural frequencies allows to identify resonance as shown on the following Figure 9. Figure 9 - Typical Campbell diagram Figure 8 - FE model and static stresses speed C. Dynamic analysis Frequency study: the Campbell diagram The first step of the dynamic analysis is to compute the modeshape and the frequencies. As presented previously, The principle of design is to identify the crossing in the operating range and to move them away by changing the blade geometry and consequently its frequencies. This is the most constraining step of the aeromechanics optimisation. Once the structure is modified, the CFD analysis must be updated to account for the impact of the geometric modifications. The aerodynamic designer proposes then a new geometry, including the previous modifications, but with some changes to restore the performance level, etc This process is called 6

7 Aeromechanics Iterations. In general 3 or 4 iterations between the aerodynamic and the stress assessment are necessary. In some particular case, up to 10 iterations may be required before the convergence. Modeshapes study In fact, after the frequency analysis, it appears very difficult, even impossible, to remove all the resonance from the operating range. The modeshapes, which are the other result of the dynamic analysis can provide interesting data. Effectively, the shape of the mode allows to classify them: bending or flexion modes, torsion modes, edgewise modes, complex modes, as presented on the following Figure 10. The empirical knowledge of engine designers allows to estimate the dangerousness of a given mode in function of its shape. This is a major assumption for design but it allows to focus the previous frequency optimisation only on the most dangerous modes to be removed from the operating range. The others crossings identified can remain but their response levels will be measured during engine testing to check they are low enough. Figure 10 - Modeshapes - 1st bending, 2nd bending, 1st torsion, complex modes (rank 15 and 16) Stress assessment The modeshape being calculated, the dynamic stresses distribution for each mode is known. The distribution only is available, not the response levels (a modal analysis only has been performed). Figure 11 - Second bending and first torsion mode dynamic stresses Each Finite Element (FE) code has its normalisation technique. The stress distribution can be expressed in such a way the maximum displacement on the model is 1mm, or the generalised mass is 1, etc... A stress distribution for a first bending mode is presented on the Figure 11. On the other hand, the static stress level is known. The dynamic stress assessment is performed using a Goodman diagram. This is a way to represent the material properties for crack propagation. A Goodman diagram gives: - on x-axis, the static stresses - on y-axis, the dynamic stresses (the amplitude 0/peak of the dynamic stress) - Different curves corresponding to the material static/dynamic stress limit before failure for one given number of cycles. Goodman curves for 10 6, 10 7, 10 8 cycles and upper are very close each over. The curve for 10 7 cycles is commonly used for the design (and is assumed to be an infinite number of cycles, as 10 7 and 10 8 curves are superposed ). If the static / dynamic stresses couple is under this curve, it means the failure will not occur in 10 million cycles, which is necessary to avoid any life limitation on the structure. The couple static/dynamic stress is intended as follow: the number of cycle is counted for the number of dynamic stresses oscillations around the static stress value. Intersections of the curves with the axes correspond to purely dynamic or purely static load cases. The following approach must be performed for each mode suspected to be excited in the operating range. For each element of the structural model, the couple static and dynamic stresses are plotted in the diagram. When all the elements are located, the objective is to determine the multiplying factor, α, applicable to dynamic stresses such as the worst couple of stress is on the modified Goodman curve and all the others are lower this curve. It is reminded that the dynamic stresses are normalised. An illustration of the use of the goodman diagram is proposed on the Figure 12. 7

8 Figure 12 - Goodman diagram illustration This approach is very rich in term of analysis: - The elements whose stresses couple is close to the curve are the most critical ones. In case of significant vibratory stresses, the area in which the blade will fail can be identified. - The multiplying factor α identified previously corresponds in fact to the maximum dynamic response allowed for each mode. If the modeshape is normalised to 1mm maximum displacement on the blade, the maximum allowable displacement regarding the Goodman diagram will be α mm. - This is also possible to multiply the dynamic stresses distribution (normalised) by α to see the maximum stresses allowable in any area of the blade. - Finally this coefficient is used for the engine testing (see the specific chapter). Concerning the Goodman diagram, it is obtained by material testing on simple sample superposing a static load and a dynamic load and counting cycle to rupture. The minimal curve must be used (-3σ). But the curves obtained can be influenced by the geometry of the blade. Moreover, the material testing is a single direction testing (static and dynamic stress in the same direction). For these reasons, the use of raw material curves on the real structure needs some update. This update is performed by a component test series, called staircase test. The blade of interest is clamped on a vibrating table and then excited on one of its modes by successive increase of dynamic levels. 1 million cycles are performed for one dynamic level, then 1 million at a higher level, etc When the structure fails, the maximum allowable dynamic stress is determined. It allows to update the whole Goodman diagram in such a way the 10 6 cycles curve passes by the previous point on the y-axis (static load is null). Dynamic model update It has been seen all along this design process that the modeshapes determination (and the consecutive dynamic stresses distribution) is a critical issue for stress assessment. It is the reason why, the airworthiness authorities ask the manufacturer to check the capability of the finite element models used for engine certification. As a consequence, specific dynamic testing on blades is performed to evaluate the accuracy of the FE model. These tests can be performed using various technologies in order to identify experimentally the dynamic stresses field on the blade surface. Figure 13 shows, from the left to the right, the studied blade, the area of measurement comparison and the modal displacements obtained by the model and the measure (holographic interferometry). Figure 13 - modal analysis: model / test comparison The measured data is projected on the finite element model in order to compare the displacements field. Result is expressed as the MAC value (Modal Assurance Criterion) which is the normalised scalar product between the measured and computed vectors. If the value is close to one, the vector are co-linear, mode shapes are similar. Figure 14 shows a MAC matrix with only the few first modes properly computed. After the fifth mode, there is no diagonal dominance indicating the model is not accurate enough. D. Engine testing Figure 14 - Example of a MAC matrix The engine test is the final step of the design process. The stresses must be measured under rotation and it must be demonstrated that the dynamic stress levels are acceptable regarding the Goodman diagram. This test is performed at the very end of the development process, many months after the components design. Blades are currently instrumented with strain gages but some others measurement techniques are under development. The gage position is optimised regarding various constraints: - the gages must not be in an area where the stresses have a strong gradient. - Some gages (3 or 4 per blade, on few blades of the bladed disk) must be redundant - The gages must not be too close each others - The gages must measure as many modes as possible with a sufficient sensitivity - Technologic constraints, due to the thermal field on turbine for instance, etc The gages are stuck on the blades and a telemetry or a slip 8

9 ring are used to transmit the signals from the rotor to the ground. During testing, the engine is accelerated and decelerated slowly in order to excite the various modes remaining in the operating range. The speed variation induces an excitation frequency variation on each harmonic of the rotation speed. The response levels are surveyed and compared to the maximum allowable limits (α coefficient previously determined). IV. Limitations There are two main limitations associated with the previous design process: - The engine test is performed at the end of the design process as shown on Figure 16. It induces that the stress level estimation is obtained a very long time after the component design. If a high response level is recorded during engine testing, the corrective action can lead to a significant over-cost and delay. - The choice of the crossings acceptable in the operating range is based on simple empirical criteria. The new blade shapes obtained thanks to the new CFD solutions make difficult the estimation of the resonance dangerousness. The modeshapes are different, some local modes appears at lower frequencies, etc and there is no experience collected for this types of geometry. Figure 15 - typical measured response under rotation The engine is certified when the dynamic stresses are lower the limit with a security margins. This margin is used: - To cover all the engine production (stresses are measured on one engine only). - To cover the uncertainties on the gage locations - To allow blade ageing inducing wear and small impacts without risking any failure. E. Conclusion The standard aeromechanical design practice has been presented from the CFD design to the engine certification tests. But of course, other constraints are applied on the bladed disk design, such as rotor / casing contacts management, blade tip clearance estimation and management, ingestion problems (water, ice, birds, ), flutter, etc Moreover, most of the excitation sources are harmonics but some non synchronous vibrations can also be identified. For these types of excitation, specific analyses are conducted at the end of the design process. The compromise between a good performance level and an acceptable dynamic situation is the corner stone of a satisfactory aeromechanical design of bladed disks. Figure 16 - Bladed disks aeromechanical design principle V. Forced response prediction The main interest of forced response computation for bladed disk is to avoid the two previous limitations: - The forced response estimation can be performed earlier in the design process in order to determine if the dynamic stresses are acceptable or not. For this point, the absolute response level is required which implies that reliable methods are available to reach the degree of accuracy of an engine testing. - The second interest of forced response computation is for the crossings dangerousness assessment. It is difficult to compute a reliable absolute response level but it is much easier to estimate the relative response between different resonances. This chapter presents the principles of forced response computation and how it is introduced in the design process in terms of relative approach. The issues related to the computation of the absolute response levels will also be detailed. 9

10 A. Principle of forced response computation The equation to solve is given hereafter. Compared to the modal analysis, the damping matrix [C] is required as well as the unsteady forces vector F(t). Elementary forces are computed as the integral of the unsteady pressures on each finite element surface. It is intended that an interpolation is required between the CFD and FE meshes which are different. Moreover, it is reminded that the CFD mesh corresponds to an "operating condition" while the FE mesh corresponds to the blade "at rest". M X! ( t) + C X!( t) + K X ( t) = F( t [ ] [ ] [ ] ) i! t Forces can be assumed harmonic: F( t) = fe. In such a case, only the first harmonic of unsteady pressures is taken into account but the formulation can be generalised to multi-harmonic cases. With X ( M ) X = f i! t ( t) = Xe,!" 2 [ ] + i" [ C] + [ K] Φ being the structure modeshapes, the solution can be expressed under the form: X=x.Φ This projection on the modal bases allows reducing the size of the problem to few modal degrees of freedom. The solution is not X, that is to say the displacement of each node of the structure (the number of unknown is the number of degrees of freedom). The solution is x, which corresponds to the participation of each mode in the response. Around one resonance, one mode dominates the response, unless in particular cases with a very high spectral density (many modes on a small frequency range). Even in that case, only 4 or 5 modes can be excited, so the problem is reduced to only 4 or 5 unknowns. With this assumption, the previous equation can be rewritten as follow: (! # 2 [ M ] + i# [ C] + [ K] ) x" = f 2 #" [ M ] + i" [ C] + [ K]!. x T ( )! f T! = T T T T ( "! [ M ]! + i"! [ C]! +! [ K]!) x =! f # 2 (! " 2 m + i" c + k) x = g The modal analysis provides k and m, but for the forced response computation, c and f must be identified. B. Remark concerning c and f: The damping can be decomposed in a mechanical and an aeroelastic term. Mechanical damping Mechanical damping is due to the fact that the material is never perfectly linear. Even for oscillating stresses lower than the yield stress ("elastic behaviour"), some energy is dissipated. Another source of mechanical damping is related to the junctions behaviours at the structure parts interface (junction blade to blade, blade to damper, blade to disk, etc ). This type of damping is associated to the non linear behaviour of the structure. An example of such an effect is presented in the chapter 6-2. Non linear damping is generally higher than the material damping which raises some problems for blisks (no interface: the bladed disk is continuous). Aeroelastic damping aeroelastic damping is related to the fluid unsteadiness generated by the blade dynamic motion in the flow. When the bladed disk vibrates, even in a perfectly homogeneous flow, it generates unsteady pressures. These unsteady pressures can either excite the structure (flutter) or damp it (in most of the cases). The aeroelastic damping depends on the flow parameters, on modeshapes, frequencies and blade phase angles (number of nodal diameters). The aeroelastic design of bladed disk is presented in another lecture series. Most of the time, the concern is to identify the risk of flutter. But the aeroelastic computation results are very interesting, even when there is no flutter risk, because it allows determining the damping required for forced response analysis. Force estimation Many sources of excitation are present in the engine but only the harmonic ones based on an aerodynamic origin will be illustrated in this document. This document will focus only on the most frequent case which is the one of the bladed disk excited by the upstream stator wake. C. numerical approaches Fluid structure interaction where m T [ M ] c T [ C] k T [ K] and g T =!!, =!!, =!! =! f are respectively the generalised masses, damping, stiffness and generalised forces (one scalar per mode). The studied problem involves a fluid-structure interaction. Forced response problems are strongly connected to aeroelasticity and various coupling techniques are commonly used. Effectively, the blade response generates unsteady pressure which can modify the response itself. 10

11 - Weak coupling: the structure and the flow behaviour are assumed linear: this is the technique used commonly in the industry because it gives a good indication of responses for a low CPU cost. The structural modes, unsteady forces and aeroelastic forces are computed separately. Aeroelastic forces are assumed linear with the blade response amplitude. The bladed disk vibration level is assumed linear with the excitation force. These parameters, identified separately are connected each over by a linearity assumption to compute the response. - Intermediate coupling - the structure is linear while the flow is not: this is the case of more complex flow fields where the aeroelastic forces can not be assumed linear with the bladed disk vibration amplitude. The forcing function and the modeshapes are estimated separately. An iterative process is then used to compute the response: the aeroelastic damping is first estimated. A response level is computed. The amplitude is used to update the aeroelastic damping estimation (non linear with the level). A new response is computed, etc - Multi-physics coupling - the structure and the flow are non linear: when the structure non linearity can be projected on a linear representation basis, the intermediate coupling can be used. Else, all the system has to be modelled: the structure response is not linear with the excitation amplitude as well as the aeroelastic damping is not linear with the vibration amplitude. In such a case, the complete modelling of the problem allows to compute at each time step, the forces applied on the blade suction and pressure side (the forcing function and the aeroelastic forces are mixed). The damping used to determine the response is the mechanical damping only. An iterative integration is required. At each step the aerodynamic forces are applied on any point of the structure, the structure displacements are computed (explicitly). The flow path being modified, the unsteady pressures are modified, etc The solution is reached when the oscillation of the blade is stabilised. Forced response computation The forced response computation process for the case of the weak coupling is performed as follows: - the structural behaviour can be computed by any commercial software (frequencies and modeshapes). - The steady and unsteady CFD can be computed by turbomachine codes - A coupling software (generally an in-house software), allows to extract the unsteady pressures harmonics from the CFD result. These results are associated to a geometry in operating condition (CFD geometry). It is then necessary to project these results on the finite element model, which is a geometry at rest. Moreover, the space discretization of the finite element model is different from the CFD mesh. A specific treatment of unsteady data is necessary. - The coupling software computes the aerodynamic generalised forces (f) - The damping value is given by the designer or is extracted from an aeroelastic computation. - The coupling software solves problem, using the FE model generalised mass and stiffness. It can be noticed that, if the problem is solved in the modal domain, the problem size is very small (1 degree of freedom if only one mode is studied). The result is a scalar, which is the amplitude of the response for the concerned mode. The CPU cost to solve the problem is consequently very low (less than 1 second). The cost associated to such an analysis is related to the pre-required finite elements and fluid dynamics computations. D. Forcing function estimation In the case of the weak coupling, different CFD techniques are used to determine the aerodynamic forcing function. It is reminded for this type of coupling, the forcing function is not affected by the structure motion: aeroelastic forces are computed separately. Typical practices are presented hereafter for a bladed disk excited by the upstream stator vanes wakes. It is clear that the airfoil shapes being designed and optimised in 3D, the unsteady computation must be performed in 3D to catch all the flow components. Steady state The excitation is due to the flow heterogeneity upstream the bladed disk. The first step is to compute the steady state, for the rotation speed of the crossing. Generally, this is performed using a full stage modelling (stator + rotor). The pressures are averaged at the outlet of the stator and the mean conditions are used as upstream boundary conditions for the rotor. This computation must be performed accounting for the fluid viscosity in order to generate the appropriate wakes downstream the stator. Only one stator passage and rotor passage is modelled. Figure 17 shows a typical steady state computation for a full stage (stator + rotor, with a mixing plan, NS 3D). The total pressure downstream the stator is shown: this will be the excitation source of the rotor. The entropy field is also presented at 90% span. The wake generated downstream the stator is not transmitted upstream the rotor (averaged in the mixing plan). 11

12 Full stage non linear unsteady Navier Stokes Figure 17 - Steady state 3D NS computation on a full stage (mixing plan) Unsteady computation: gust technique The objective of this technique is to avoid a viscous unsteady computation on a full stage, which is very expensive. This technique consists in the extraction of the steady flow parameters (from the previous steady computation), downstream the stator. These parameters are then introduced as upstream boundary conditions (not averaged this time) in an Euler unsteady computation on the rotor only. In such a case, only one interblade passage is modelled and chorochronic boundary conditions are used. Effectively, the number of stator vanes is different from the rotor blades number. The space and time symmetry is not the same for the stator and the rotor. A specific treatment is performed on the steady parameters to fit with the rotor symmetry. The main limitations of such a technique are: - the geometric location of the plan where the steady parameters are extracted and re-injected in the Euler computation has a first order influence of the result. - The unsteady computation performed with an Euler formulation on the rotor is attenuating the viscous effect. The wake is mixed in the flow and its intensity decreases. This contributes generally to underestimate the response levels. Full stage unsteady linear Navier Stokes Another solution consist in modelling a full stage in 3D (1 stator passage and 1 rotor passage with chorochronicity), with a Navier Stokes linear formulation. It allows accounting for viscous effect but the flow behaviour is assumed not to change much around the steady state solution. The interest of such an approach is to model the viscosity at low cost for a full stage computation. There is no sensitivity to the extraction plan downstream the stator like for the previous gust technique. The main limitation of this technique is the linear formulation which does not apply systematically. Indeed, the unsteady flow structure can be significantly different from the steady one and this kind of approach do not fit necessarily. This technique is much more expensive but requires less preliminary assumption. It is used for design check or in service problems but it is often too heavy to be used during aeromechanical iterations. The remaining assumptions are associated to the turbulence formulation, which are discussed in many publications. Figure 18 shows a typical unsteady computation result with a full stage, NS 3D approach. The entropy field at 90% span is shown. Compared to the steady state presented on the Figure 17, the wake is transmitted to the rotor domain (no averaging between the stator and the rotor). These computational results are used to determine the unsteady pressure fields on the airfoil suction and pressure sides (see Figure 19). Figure 18 - Full stage unsteady NS (3D) computation Figure 19 - unsteady pressures first harmonic on the rotor blade surfaces E.Forced response in the design process Forced response computation can be used in the aeromechanical iteration in the following ways. First iterations: The forced response computation can be made on few crossings of the Campbell diagram. In a first approach, the idea is to determine which crossing has the higher response and to focus the dynamic optimisation on these crossings only. In such a case, the blade alone can be used to estimate the response level and a simplified CFD computation (gust technique or linearised NS) is required. 12

13 At this stage, a relative approach between various crossings is sufficient. 1- a Campbell diagram is plotted without cyclic symmetry (the disk is not perfectly defined yet). 2- the rotation speeds for the five or ten first modes crossing in the operating range are identified. 3- the forcing function is estimated in function of each rotation speed. In order not to multiply the number of CFD steady computations, only few speeds can be computed (if some crossings are closed for instance) amongst the 10 identified previously. 4- the forced response is computed by a specific software. - If significant response levels are identified during engine testing, it could be interesting to assess the measured results dispersion. Forced response computation can be used to generalise a measurement. Figure 20 shows a typical response in term of maximum displacement on the blade within a frequency range. This type of result is of first interest for the designer. Last iterations The idea is to check the design choices and to determine if the response levels are acceptable regarding the Goodman diagram, before the engine testing. In such a case, more accurate results are required. The analysis is consequently more expensive, but it can be focussed on few crossings only. A more precise CFD analysis is necessary (full stage unsteady NS) and the bladed disk modes must be computed with the cyclic symmetry assumption (and updated with component tests). 1- the 2 or 3 concerned crossing are identified 2- an accurate FE modelling is made, of one sector (blade + disk) with the boundary condition consistent with the order of excitation and the rotation speed associated to the 3 previous crossings. 3- step 2 results are used to update the Campbell diagram and estimate precisely the speed of the crossings. 4- CFD computation, steady state and unsteady for each crossing 5- response computation by the coupling software 6- assessment of the response levels regarding the Goodman diagram and risk evaluation. Engine testing or in service analyses When the engine testing has been performed (or after entry into service), many experimental data are generally available. Forced response computation can be used to understand problems or to extrapolate results. - for instance, during the engine testing, some crossings can not be studied (effect of ageing on frequencies, ground testing limitation, etc ). In such cases, forced response computation can allow to extrapolate the missing experimental results. - After engine testing, small configuration change could influence the dynamic behaviour of bladed disk. Based on the engine testing, the model can be updated to assess the response levels on a new configuration. F.Conclusion Figure 20 - Typical forced response computation result Forced response computation can be performed with various degrees of accuracy. Depending of the step in the development process, global trends as well as precise results can be useful. The main objective is to reduce the High Cycle Fatigue (HCF) risk and to extrapolate one engine testing to many configurations or operating conditions. Relative approaches are interesting to classify the resonance dangerousness. But accurate results are sometimes necessary, and in that case, forced response computation can be very complex: - An accurate result requires a high quality unsteady CFD analysis which is very expensive. - But it requires also accounting for some mechanical particularities that can influence much the response level. The main influencing parameters are the mistuning and damping, always present in the engine but much more complex to estimate. The estimation of damping in particular is a weak point of the analysis: this topic is clearly an issue for which improvements are required. A synthetic illustration is proposed in the following chapter. 13

14 VI Specific aspects related to bladed disks Two main aspects are presented shortly in this chapter. - Mistuning can influence much the response level: the response on each blade may be very different and some blade responses can double compared to the tuned case. - Friction damping, due to under-platform dampers commonly used in turbomachines, can multiply by a factor 2 or 3 the damping of the structure, thus reduce in similar amplitude the response levels. A. Mistuning Definition Up to now in this document, the bladed disk was supposed to be cyclically symmetrical structures. In fact, each blade is different from the others, due to manufacturing tolerances, mounting clearances, material characteristics dispersion, etc These small variations of blades characteristics imply that each blade dynamic behaviour is different from the others: the frequency scatter can reach one or two percents standard deviations: this frequency scatter is called mistuning. This small scatter is sufficient to break the structure symmetry and the effects are huge. Indeed, in cyclic symmetry, the dynamic waves rotation on the bladed disk was the guaranty of a good circulation and distribution of the vibration energy on each sector of the bladed disk. In the mistuned case, each blade response is different from the others and there is no constant phase angle between the sectors. The energy can localise on only few sectors, generating important responses on few blades. As an illustration, on a bladed disk with N blades (N pair to simplify), for each mode family (first bending, first torsion, second bending, etc ): - in the symmetric case, each mode family was composed by 2 real modes (0 and N/2 nodal diameters) and (N/2)-1 double (complex) modes, associated to (N/2)+1 modeshapes corresponding to the various number of nodal diameters. - In the mistuned case for each mode family, there are N different modes without any particular logic (no constant phase angle, not the same amplitudes). It is important to notice that the disk loss of symmetry do not affect significantly the whole response of the bladed disk. The blade mistuning is clearly the dominant parameter. Nevertheless, the coupling between the mistuning levels and the disk stiffness drives the response level. If there is only a poor dynamic coupling between the blade and the disk, the mistuning would not have a significant effect. Numerical approaches It is obvious that the structure not being symmetric, its representation by a one sector modelling only is not sufficient anymore. Various numerical approaches are proposed to compute the mistuned forced response: - the full 3D representation (360 ) of the bladed disk can be used but is very expensive. Moreover, the introduction of mistuning is not easy because the geometric or material characteristics dispersion must be determined. - A simple technique consists in the FE mesh simplification or reduction by typical condensation techniques such as super-elements. It allows reducing the size of the model but the basic geometric perturbation must be known. - More adapted techniques have been developed during the last years to account for mistuning with accuracy and reduced computational costs. A typical technique, based on the branch method of Benfield and Rhuda is described in reference [1]. As for the cyclic symmetry, the problem is solved in generalised parameters in order to reduce the computational costs. The modal projection basis is richer than for cyclic symmetry: it is based on disk modes (computed with cyclic symmetry) and blade mode. The mistuning is applied by perturbing the generalised blade matrices (mass and stiffness). Different simulation techniques are available with various levels of accuracy and cost. Nevertheless, another major difficulty is related to the mistuning identification. Each blade frequencies can be identified before mounting on the disk for a specific test (engine testing for instance). But if the objective is to generalise this test results to a complete fleet of engine, the mistuning patterns associated to this fleet are not known. In order to assess the design robustness to mistuning, statistical approaches are often required. This is the reason why, cost reduction for mistuning analysis is a key point, mainly in probabilistic approaches were a significant amount of configurations must be computed. Typical responses The typical response of mistuned bladed disks can be presented as the amplification of the tuned response. The tuned response can be computed as presented previously in this document, and the amplification factor is then estimated for each blade. A typical response is proposed on Figure 21. It can be seen that the blade to blade response can vary much: a factor 10 scatter is possible between the maximum response blade and the minimum response one 14

15 (only 4 on the following figure). The maximum response on one blade can be more than twice the tuned response value (1.5 on the following figure). It can also be seen that the mistuned response level depends on the mistuning standard deviation and the blades distribution around the disk. One set of blades arranged differently around the disk can lead to completely different dynamic responses. Figure 22 shows also that the response amplification do not increase with the mistuning standard deviation. Values on this figure are of course indicative: they depends of the mistuning itself but also of the coupling between blades (disk / blades relative stiffness) and of the damping. Friction damping in bladed disks is generated by the interfaces between the various components of the engine. The optimisation of interfaces can provide extra damping to the structure and new interfaces can be introduced in the turbine to increase the damping value, as for instance (see Figure 23): - the shrouded blades contact area and pre-stresses can be optimised regarding damping - some dampers can be introduced under the blades platforms to generate and extra interface - some damping rings can also be used inside the rotor (between disks for instance) Figure 23 - Various interfaces between blades and disk This paragraph object is to present the behaviour of friction dampers and how they can be optimised. Physical behaviour Figure 21 - Response distribution on each blade within a frequency range Max Amplification 3-1, Mistuning standard deviation (%) Figure 22 - Maximum amplification in function of the mistuning standard deviation B.Friction damping Definition As shown previously, damping is a key point regarding forced response. Obviously, it is of first interest to maximise damping in order to reduce forced response. Damping has various origins (aeroelastic, material non linearity) but the predominant one is the damping at the structure interfaces. To illustrate this, the single piece bladed disk (so called blisk) has a significantly lower total damping than a classic bladed disks. Damping at the structure interfaces is not due to the thermal dissipation associated to friction. In the case on underplatform dampers, their mass is only few grams and the energy they could evacuate thermally is very small. The damping origin is due in fact to the structure non linearity. The friction damper is stuck under the blade by the normal forces due to rotation speed. It can start to slip if the tangential forces due to the blade vibration are sufficiently strong, i.e. greater than the friction coefficient multiplied by the normal force (see Figure 24). For one given rotation speed crossing, the mass of the damper can be optimised in such a way the damper alternates stick/slip state during the vibration: - No crossing, no dynamic level, the damper is stuck: this state corresponds to an added coupling stiffness between the two blades. - When the crossing conditions are close, the response levels are increasing. The dampers start to slip. When they slip, the extra coupling stiffness added by the damper disappears, generating the blade frequencies decrease. The mode is not excited anymore and the dampers sticks again. The system alternates two states and it is not possible to establish the forced response regime. The apparent damping is consequently due to the system non-linearity. 15

16 To be efficient, dampers must respect the following criteria: - not be too light regarding the excitation level, else the damper always slips: no interest - not be too heavy regarding the excitation level, else the damper is always stuck: no interest - be stiff enough in order to separate significantly the natural frequencies in stuck or slipping condition. - the dampers interfaces with the blade must be located in an area were the modal displacement, tangential to the contact surface, are significant for the dynamic response to damp. In other words, the mode to damp must have significant displacement in the contact region. The contact surface can be optimised (geometry, local modeshapes, friction coefficient, etc ). If the dampers is well designed for the mode of concern, it will alternate sticking and slipping phases at the resonance, introducing a significant non linearity in the system. The natural frequencies of blades are always changing: it is then impossible to reach a full resonant configuration. This contribute in reducing the response level, providing a significant apparent damping. Figure 24 - Damper location and applied forces Numerical approaches The forced response simulation with damping implies necessarily a non linear approach for structural mechanics. Two types of numerical approaches can be used: - non linear time history analysis for the structure. This is a simple but very expensive computation. Indeed, the stabilisation of the damper behaviour (stick/slip transitions) implies to compute a significant amount of vibration periods. The physical time to be computed is long regarding the very local non linearity management (small convergence time steps to compute a long time phenomenon). - Multi-harmonic balance method: The structure is reduced on a modal basis, composed mainly by the blade modes with the dampers stuck and the blade modes without dampers. It is assumed that any response of the blade will be a combination of these two types of basic modes. The blade is reduced as a superelement (Craig-Bampton type for instance) on its non linear degrees of freedom, that is to say the interface points with the damper. The cost to determine the blade motion is consequently extremely reduced and depends only the non linear behaviour of the damper. The damper behaviour is computed thanks to a time history integration. The harmonic balance method is an iterative process to exchange information between the modal domain (blade) and the time domain (damper) and insure the contact laws are respected at the interface. The equation to solve is:! " 2 m + i" c + k x + Fnl = ( ) g The philosophy of the harmonic balance method can be summarised as follows: - The damper is supposed stuck (F nl =0) - The forced response of the structure (linear) is computed in the modal domain - An inverse Fourrier transform allows to built the time history of the non linear degrees of freedom (contact nodes between the blade and the damper) - A time integration of the non linear problem is performed. The input is the imposed motion of the non linear degrees of freedom. The damper behaviour is computed (stick/slip transitions) and the non linear force applied under the platform is identified. - A Fourrier transform of the non linear force time history is performed. A significant amount of harmonic should be use for a good description of the non linear force in the modal domain. - The non linear force Fourrier terms are used to compute a new forced response (multiharmonic response) of the blade in the modal domain. - An inverse Fourrier transform, converts the blade multiharmonic response in the time domain - Etc The iterative process stops when the non linear force estimation converges. The interest of such an approach is the possibility to capture non linear phenomenon by a multiharmonic computation. The non linear problem is reduced to few non linear degrees of freedom. This technique is very efficient in terms of computational costs. Typical responses A typical response is shown on the following Figure 25. The blade vibratory response within a frequency range is given in function of the damper mass (for one given friction coefficient). The frequency scatter between the two peaks is due to the added stiffness of the damper. The 16