Optimization-Aided Forced Response Analysis of a Mistuned Compressor Blisk

An axial turbine blisk for turbocharger applications is analyzed with respect to the effect of intentional mistuning on the forced response. Originally, the intentional mistuning pattern has been designed by employing a genetic algorithm optimization in order to reduce the forced response caused by low engine order excitation (LEO) of the fundamental flap mode. The solution found has been implemented in a prototype of that blisk. For the purpose of comparison, a second reference blisk has been manufactured without intentional mistuning. The actual mistuning distributions of the blisks have been identified by employing blade-by-blade impact testing. Alternatively, a new inverse approach has been employed, which is based on a least squares formulation and benefits from less experimental effort. Based on the information gained by the aforementioned testing procedures, subset of nominal systems (SNM)-models have been updated, which allow for considering the aeroelastic coupling by means of aerodynamic influence coefficients (AIC). Despite of small but unavoidable deviations from the design intention it could be proved within numerical simulations that the intended 70 per cent reduction of the maximum forced response is nevertheless achieved. In addition, the paper is addressing the effect of the aforementioned intentional mistuning pattern on a higher mode, which is relevant for the durability as well. Hence, new SNM-models have to be updated in order to calculate the forced response due to EO-excitation caused by the nozzle guide vane. Although the original mistuning pattern has been optimized solely for reducing the forced response of the fundamental flap mode, it hardly affects the higher mode forced response in a negative manner.

The introduction of mistuning in jet-engine bladed disks can lead to large changes in stability and forced response. Even small random mistuning (within the bounds of manufacturing tolerance) can lead to unacceptable response and high-cycle fatigue. Meanwhile, intentional mistuning may improve stability and forced response under manufacturing uncertainty. This paper presents a general framework for predicting forced response as a function of mistuning. Because the forced response problem is an almostsingular linear problem, its solution is highly nonlinear in the mistuning parameters. Our methods exploit symmetry arguments and eigenstructure perturbation to provide a method valid for any model. It is shown that, by perturbing eigenvectors in the numerator and the inverse of eigenvalues in the denominator (exploiting symmetry in both computations), we can accurately approximate the forced response as a function of mistuning. Results are demonstrated for a simple lightly damped model, and the consequent sharp nonlinear behavior is captured almost perfectly. We also show that intentional mistuning may guarantee improved stability and forced response under fixed manufacturing tolerances. Thus, intentional mistuning should be considered as a practical means of increasing safety and enhancing engine performance.

α = magnitude of the change in design variables β = new design variable (Vanderplaats’ formulation) e = relative error λ = buckling load factor σ = normal stress c = positive constant (Vanderplaats’ formulation) F = objective function g = design constraint i = design variable index j = active constraint index p = parameter ∆p = parameter increase r = inactive constraint index S = vector of feasible directions w = vertical displacement X = vector of design variables X = vector of optimum design variables Xaprox = vector of approximate optimum design variables

Intentional mistuning is the deliberate incorporation of blade-to-blade parameter variations in the nominal design of a bladed disk. Previous studies have shown that this is a promising strategy for mitigating the damaging effects of unintended, random mistuning. In this paper, the mechanisms of intentional mistuning are studied by investigating the relation between blade response and vibration energy flow in lumped parameter models. Based on key observations from the energy flow analysis, a few design guidelines are proposed that drastically reduce the design space for intentional mistuning patterns. Thus, an optimization may be performed on the reduced design space or skipped altogether, yielding dramatic reductions in computational costs. The guidelines are validated by extensive Monte Carlo simulations for the lumped parameter models as well as for a finite-element-based reduced-order model of an industrial rotor. It is shown that the reduced design space includes optimal or near-optimal intentional mistuning patterns.

The effect of intentional mistuning has been analyzed for an axial turbocharger blisk with the objective of limiting the forced response due to low engine order excitation (LEO). The idea behind the approach was to increase the aerodynamic damping for the most critical fundamental mode in a way that a safe operation is ensured without severely losing aerodynamic performance. Apart from alternate mistuning, a more effective mistuning pattern is investigated, which has been derived by means of optimization employing genetic algorithms. In order to keep the manufacturing effort as small as possible, only two blade different geometries have been allowed, which means that an integer optimization problem has been formulated. Two blisk prototypes have been manufactured for purpose of demonstrating the benefit of the intentional mistuning pattern identified in this way: A first one with and a second one without employing intentional mistuning. The real mistuning of the prototypes has been experimentally identified. It is shown that the benefit regarding the forced response reduction is retained in spite of the negative impact of unavoidable additional mistuning due to the manufacturing process. Independently, further analyzes have been focused on the robustness of the solution by considering increasing random structural mistuning and aerodynamic mistuning as well. The latter one has been modeled by means of varying aerodynamic influence coefficients (AIC) as part of Monte Carlo simulations. Reduced order models have been employed for these purposes.

A preliminary inviscid version of the NASA LeRC/Mississippi State University 3-D aeroelastic computational fluid dynamics code, TURBO-AE, is evaluated using operating conditions from three test cases from an AlliedSignal Engines (AE) advanced technology integrally bladed fan rotor. Two of the cases corresponded to conditions where high op-line flutter vibrations were observed, and the third case was flutter-free during testing. TURBO-AE steady (no blade vibration) computations were performed for each test case and a reasonable match was found with the steady 3-D viscous code DAWES. Following completion of the steady computations, TURBO-AE flutter (blade vibration) analyses were completed by harmonically oscillating the airfoil with mode shapes computed by the structural finite-element code ANSYS. Aerodynamic work per vibration cycle and modal damping was computed by TURBO-AE at 0° and 180° Inter Blade Phase Angles (IBPA). For IBPA = 0°, TURBO-AE predicted aerodynamic damping to be low but positive. Further, for IBPA = 180°, the TURBO-AE predictions indicate a very high level of aerodynamic damping. Flutter was observed on the rotor only for IBPA = 32.7°, and TURBO-AE analyses are in process for this condition. Unsteady pressure perturbation contour plots are presented to aid in the visualization of the TURBOAE flow solution. Results presented in this paper indicate that, although TURBO-AE still requires significant development, it is making the difficult transition from a research code to a flutter and forced-response prediction tool for the U.S. aeropropulsion community.

This study aims to numerically investigate the sensitivity of the forced response with respect to the variation of the tip clearance setting of a low pressure compressor BluM™(monoblock bladed drum) when it is subjected to low engine order excitations. Two different types of blades are employed in the upstream row in order to generate the low engine order excitations. The forced response as well as the aerodynamic damping is numerically estimated using the TWIN approach. The experiments are conducted to measure the forced response for the nominal tip gap to validate the numerical results. Further, simulations are performed for a range of tip clearances. The variation of the steady load distributions due to the changes of the tip clearance are analyzed and presented. The aerodynamic damping and the forced response are calculated and compared for different tip clearances. It is observed that aerodynamic damping increases significantly with tip gap, whereas the excitation forces are reduced. As consequence of these two evolutions, the forced response decreases drastically for larger tip clearance.

In this article the effect intentional mistuning of an axial turbomachine bladed disk has been analyzed in order to reduce forced response due to low-order engine excitation. The maximum value of forced response of turbomachine rotor’s blades with mistuning parameters is usually much more than the value of the tuned rotors. An increase level mistuning of this critical value actually leads to a decrease magnifications of the forced response. Thus, the actual work has been introducing some degree of intentional mistuning in the design to achieve these purposes. The effectiveness of intentional mistuning has been researched at the design stage of the bladed disk in the energy turbomachines, which is introduced into the rotor’s design by changing the nominal mass of the blades in harmonic models.

Rechnergestützte optimierung der antriebs- und gestellkraftgrössen ebener koppelgetriebe—Teil II

This study is centered on the aeroelastic problem for axial compressors blisk airfoils in presence of geometrical variability. The combined problem of structural dynamics and unsteady aerodynamics is of interest for these machines due to the stress induced by blade vibration. The geometrical variability resulting from the manufacturing process of blisk airfoils and its impact on the aeroelastic problem is investigated, focusing on the aerodynamic damping. The manufacturing geometrical variability is analyzed in a probabilistic manner. A digital twin of the variability is created starting from a dataset of optical surface scans. The measured geometries are parameterized to describe the differences from the nominal geometry. An Autoencoder represents these deviations within a required accuracy, while using a minimal set of variables. The data reduction provided by the Autoencoder technique proved to be very efficient, especially if compared to linear methods as the principal components analysis, allowing to investigate multi-passage deviations from the nominal geometry. The aerodynamic damping is computed by an unsteady CFD solver, using the aerodynamic influence coefficients method. The manufacturing geometrical variability model is used to represent the real blisk airfoil geometries investigated. The uncertainty quantification evaluates the impact of the deviations from the nominal cyclic symmetrical design on the aerodynamic damping curve. The results can be combined in an aeroelastic reduced order model with the mistuning of the mechanical properties to represent the mistuned blisk vibrations.

An efficient methodology for predicting the nonlinear forced vibration response of a turbine engine rotor with a cracked blade is presented and used to investigate the effects of the damage on the forced response. The effects of small, random blade-to-blade differences (mistuning) and rotation on the forced response are also considered. Starting with a finite element model, a hybrid-interface method of Component Mode Synthesis (CMS) is employed to generate a reduced-order model (ROM). The crack surfaces are retained as physical degrees of freedom in the ROM so that the forces due to contact interaction in three-dimensional space can be properly calculated. The resulting nonlinear equations of steady-state motion are solved by applying an alternating frequency/time-domain method, which is much more computationally efficient than traditional time integration. Using this reduced-order modeling and analysis framework, the effects of the cracked blade on the system response are investigated for various mistuning levels and rotation speeds. First, the advantages of the selected hybrid-interface CMS method are discussed and demonstrated. Then, the resonant frequency shift associated with the stiffness loss due to the crack, as well as vibration localization about the cracked blade are thoroughly investigated. In addition, the results of the nonlinear ROMs are compared to those obtained with linear ROMs as well as blade-alone ROMs. It is shown that several key system vibration characteristics are not captured by the simpler models, but that some insight into the system response can be gained from the blade-alone response predictions. Furthermore, it is demonstrated that while the effects of the crack often appear similar those of mistuning, differences between the effects of mistuning and damage can be discerned by observing and comparing the response across different families of system modes.

Intentional mistuning (IM) has turned out to be an effective measure to alleviate the maximum forced response of bladed wheels in the framework of numerous studies in the past. In particular, solutions based on two different blade designs, following, e.g., alternating or AABB patterns, have proved to be promising in this regard and moreover robust against the impact of unavoidable random mistuning. Thus, for example, a 40% reduction of the first blade bending maximum forced response has been proved experimentally for a turbine impeller of a turbocharger application. Despite this success, the technical implementation of the frequency-based mistuning pattern followed an academic solution based on locally removing material at the leading edge tip, which is not suited for the use in serial wheels since it may disturb the flow channel. In addition, the forced response of other blade modes may be affected in a negative manner. In order to overcome these problems, an alternative way of implementing intentional mistuning is suggested by applying a marginal geometric modification of the blade thickness distribution to adjust the natural frequency of the first bending mode. Finite element-based shape optimization is utilized to this end. Secondary conditions are ensuring that only the target frequency of the first bending mode is adjusted, whereas natural frequencies of other modes are kept almost unchanged.

This is part two of a two-part paper. Part One describes the methodologies of a blade forced response prediction system. The emphasis of this part is to demonstrate the capability and computational efficiency of the system for predicting blade forced response. Part two firstly presents verification of the multistage time-linearized unsteady flow solver through comparison of predicted blade surface pressure distributions with data measured on a VKI transonic turbine stage. It concludes with presentation of the results of an analysis carried out on the last stage rotor blade of an ALSTOM three-stage transonic test compressor. In the analysis, strain gauge results together with Finite Element (FE) modal analysis identify the resonant crossings. The mode shape of the blade vibration is used in the CFD code to predict the blade aerodynamic damping. The aerodynamic damping is compared with the blade system damping obtained from the strain gauge tests. The variation is shown of aerodynamic and mechanical damping with blade mode shape. The blade unsteady modal forces induced by the upstream stators are derived from the calculated unsteady flows. The blade vibration at three resonant crossings is compared with those given by strain gauge measurements. Good comparisons and high computational efficiency demonstrate that the forced response methodologies described in Part One can be used in the blade design process to tackle blade aeromechanical issues.

This paper presents an adjoint analysis for 3d unsteady viscous flows aimed at the calculation of linear worksum sensitivities involved in turbomachinery forced response predictions. The worksum values are normally obtained from linear harmonic flow calculations but can also be computed using the solution to the adjoint of the linear harmonic flow equations. The adjoint method has a clear advantage over the linear approach if used within a rotor forced vibration minimization procedure which requires the structural response to a large number of different flow excitation sources characterized by a unique frequency and inter-blade phase angle. Whereas the linear approach requires a number of linear flow calculations at least equal to the number of excitation sources, the adjoint method reduces this cost to a single adjoint solution for each structural mode of rotor response. A practical example is given to illustrate the dramatic computational saving associated with the adjoint approach.

Experimental investigations of the aerodynamic damping of compressor blades are usually performed by forcing the blades externally to a motion that is similar to a given mode shape and inter-blade phase angle (IBPA) while measuring the motion-induced unsteady pressure distribution. Evaluating this to an aerodynamic work entry from the fluid to the blade, at least a local contribution to the flutter (in-)stability can be determined. Test rigs are usually non-rotating linear or rotating annular cascade wind tunnels. In the latter case, besides measuring in and transmitting data out of the rotating system, the excitation of the blades themselves is still a challenge. In the present case a blisk rotor with realistic fan blade geometries and flow conditions was investigated aeroelastically. For the excitation of the 1st bending (1F) mode shape of the blading a sector of five blades was chosen. In this sector the natural vibration of the blading, represented by constant vibration amplitudes and a given IBPA should be simulated. Therefore the blades were equipped with Macro Fiber Composites (MFC). These foils of piezoelectric material expand and contract due to the applied high voltage. A control system was developed to adjust the amplitude and the IBPA of the blade vibration. For the transmission of the high voltage, a separate data transmission system on the base of liquid metals was chosen. The blade vibration was measured by strain gauges and additionally monitored by a specific rig system. The aeroelastic investigations were carried out in the compressor test facility M2VP of the DLR Institute of Propulsion Technology in Cologne. During the measurement, the MFCs were able to excite the blades to a certain extent. The paper will present the technique to excite the blades of a compressor blisk by means of MFCs as well as achieved vibration amplitudes and limitations under engine-like operating conditions.