Analysis of Mistuned Blade Vibrations Based on Normally Distributed Blade Individual Natural Frequencies

With increasing demands for reliability of modern turbomachinery blades the quantification of uncertainty and its impact on the designed product has become an important part of the development process. This paper aims to contribute to an improved approximation of expected vibration amplitudes of a mistuned rotor assembly under certain assumptions on the probability distribution of the blade’s natural frequencies. A previously widely used lumped mass model is employed to represent the vibrational behavior of a cyclic symmetric structure. Aerodynamic coupling of the blades is considered based on the concept of influence coefficients leading to individual damping of the traveling wave modes. The natural frequencies of individual rotor blades are assumed to be normal distributed and the required variance could be estimated due to experiences with the applied manufacturing process. Under these conditions it is possible to derive the probability distribution of the off-diagonal terms in the mistuned equations of motions, that are responsible for the coupling of different circumferential modes. Knowing these distributions recent limits on the maximum attainable mistuned vibration amplitude are improved. The improvement is achieved due to the fact, that the maximum amplification depends on the mistuning strength. This improved limit can be used in the development process, as it could partly replace probabilistic studies with surrogate models of reduced order. The obtained results are verified with numerical simulations of the underlying structural model with random mistuning patterns based on a normal distribution of individual blade frequencies.

Forced Response Sensitivity of a Mistuned Rotor From an Embedded Compressor Stage

The frequency mistuning that occurs due to manufacturing variations and wear and tear of the blades can have a significant effect on the flutter and forced response behavior of a blade row. Similarly, asymmetries in the aerodynamic or excitation forces can tremendously affect the blade responses. When conducting CFD simulations, all blades are assumed to be tuned (i.e. to have the same natural frequency) and the aerodynamic forces are assumed to be the same on each blade except for a shift in interblade phase angle. The blades are thus predicted to vibrate at the same amplitude. However, when the system is mistuned or when asymmetries are present, some blades can vibrate with a much higher amplitude than the tuned, symmetric system. In this research, we first conduct a deterministic forced response analysis of a mistuned rotor and compare the results to experimental data from a compressor rig. It is shown that tuned CFD results cannot be compared directly with experimental data because of the impact of frequency mistuning on forced response predictions. Moreover, the individual impact of frequency, aerodynamic, and forcing function perturbations on the predictions is assessed, leading to the conclusion that a mistuned system has to be studied probabilistically. Finally, all perturbations are combined and Monte-Carlo simulations are conducted to obtain the range of blade response amplitudes that a designer could expect.

Effect of Mistuning and Damping on the Forced Response of a Compressor Blisk Rotor

The forced response of an E3E-type high pressure compressor blisk front rotor is analyzed with regard to intentional mistuning and its robustness towards additional random mistuning. Both a chosen alternating mistuning pattern and artificial mistuning patterns optimized concerning the forced response are considered. Focusing on three different blade modes, subset of nominal system mode-based reduced order models are employed to compute the forced response. The disk remains unchanged while the Young’s modulus of each blade is used to define the particular mistuning pattern. The well established aerodynamic influence coefficient technique is employed to model aeroelastic coupling and hence to consider the strongly mode- and inter blade phase angle-dependent aerodynamic damping contribution. It has been found that a reduction of the maximum forced response beyond that of the tuned reference can be achieved for particular mistuning patterns and all modes considered. This implies an exciting engine order which would cause a low nodal diameter mode in case of a tuned blisk. At best a nearly 50% reduction of maximum response magnitudes is computed for the fundamental bending mode and large mistuning. The solution proved to be robust towards additional random mistuning of reasonable magnitude, which is of particular interest with regard to a potential technical realization. In case of small mistuning as assumed for the first torsion and the longitudinal bending mode the advantage of achieving response magnitudes beyond the tuned reference gets lost indeed, if random mistuning is superimposed. However, mostly a lower response level is calculated compared to responses obtained from models adjusted to mistuning determined by experiment.

Estimation of Burner-Can Induced Excitation Response of an Industrial Low Pressure Turbine Blade: Validation Against Prototype Test Data

Discovering vibration problems after engine prototype tests translates in increased costs and limited re-design space. For this reason numerical tools are more and more being applied to help taking correct decision in the early design. This paper presents the estimation of the vibration levels induced by hot-streaks from burner-cans in the last blade of an industrial gas turbine. The aerodynamic forcing was obtained from full wheel time marching unsteady computational fluid dynamics calculation. The results are compared with a previous calculation of a scaled sector model. Determination of different sources of damping is achieved by the use of an unsteady aerodynamics harmonic solver and a friction damping harmonic balance method. Results show that the fatigue risk due to the first burner-can harmonic could be predicted with a fairly good accuracy from both scaled and full wheel model when compared with the strain gage prototype test data. The presence of various engine orders in the test spectral maps are commented and related to the calculations.

Resonant Response and Random Response Analysis of Mistuned Bladed Disk Consisting of Directionally Solidified Blade

Recently, DS (Directionally Solidified) and SC (Single Crystal) alloys have been widely applied for gas turbine blades instead of CC (Conventional Casting) alloys, in order to improve the creep rupture strength. The DS blade consists of several columnar grains of SC, where the growing direction of the columnar crystal is set to the direction of the centrifugal force. Because the elastic constants of the DS blade are anisotropic, the mistuning characteristics of the bladed disk consisting of the DS blades seem to be different from those of the CC blade. In this study, the resonant response and random response analysis of mistuned bladed disks consisting of the DS blades are carried out, considering the deviations of the elastic constants and the crystal angle of the DS blade. The FMM (Fundamental Mistuning Model) and the conventional modal analysis method are used to analyze the vibration response of the mistuned bladed disk. The maximum resonant response and random response of the mistuned bladed disk consisting of the DS blades are estimated by the Monte Carlo simulation combining with the response surface method. These calculated results for the DS blades are compared with those of the CC blades. From these results, it is concluded that the maximum response of the mistuned bladed disk consisting of the DS blades is the nearly same as that of the CC blades. However, in the design of the tuned blade, where the blade resonance should be avoided, it is necessary to consider that the range of the resonant frequency of the DS blade becomes wider than that of the CC blade.

Sensitivity Analysis to Flutter for Front Stages Compressor Blades

Heavy duty gas turbine front stages compressor blades aero-elastic behavior is deeply analyzed and investigated by means of an uncoupled, non-linear and time-accurate CFD URANS solver. The travelling-wave approach and the energy method have been applied in order to assess the aerodynamic damping (in terms of logarithmic decrement) for each inter blade phase angle (IBPA) and thus to localize the flutter stability region. The work is mainly focused on a sensitivity analysis with respect to blade operating conditions, eigen-mode shapes and frequency in order to improve the understanding of flutter mechanism and to identify the key parameters. Transonic, supercritical and subsonic blades are investigated at different operating conditions with their corresponding eigenmode and eigen-frequency (first and second flexural mode and first torsional). The results show that non-linear effects can be neglected for subsonic blades. Besides, the modal-shape and the shock structure, if any, are identified to play a key role for flutter stability.

Effects of Blade Count Ratio on Aerodynamic Forcing and Mode Excitability

The effects of blade count ratio (BCR) on both the steady and unsteady blade loading and the sensitivity of generalized force to a change in mode shape (mode excitability) are studied numerically on two 2D configurations: a subsonic research compressor stage and a turbine stage with supersonic exit. Using the Harmonic Balance method, only a single passage is modeled to represent the actual blade count in a row at a high level of computational efficiency. BCR variation is achieved by scaling the downstream airfoils with a fixed chord-to-pitch ratio, thus preserving the steady-state aerodynamics. It is found that the interaction among potential-, wake-, and shock-related excitations, and the relative strength of harmonic contents are dependent on BCR, resulting in a non-monotonic correlation between unsteady loading and BCR in the downstream row. It is also found that the mode excitability can be sensitive to BCR variation in both up- and downstream rows in some cases. To the best of authors’ knowledge, this is the first work on BCR study involving supersonic flow and a discussion of mode excitability patterns.

The Low Reduced Frequency Limit of Vibrating Airfoils: Part II — Numerical Experiments

This paper studies the unsteady aerodynamics of vibrating airfoils in the low reduced frequency regime with special emphasis in its impact on the scaling of the work per cycle curves using an asymptotic approach (Part I) and numerical simulations. A perturbation analysis of the linearized Navier-Stokes equations at low reduced frequency is presented and some conclusions are drawn (Part I of the corresponding paper). The first important result is that the loading of the airfoil plays an essential role in the trends of the phase and modulus of the unsteady pressure field caused by the vibration of the airfoil. For lightly loaded airfoils the unsteady pressure and the influence coefficients scale linearly with the reduced frequency whereas the phase departs from π/2 and changes linearly with the reduced frequency. As a consequence the work-per-cycle is proportional to the reduced frequency for any inter-blade phase angle and it is independent of its sign. For highly loaded airfoils the unsteady pressure modulus is fairly constant exhibiting only a small correction with the reduced frequency, while the phase departs from zero varies linearly with it. In this case only the mean value of the work-per-cycle scales linearly with the reduced frequency. This behavior is independent of the geometry of the airfoil and in first approximation of the mode-shape. For symmetric cascades the work-per-cycle scales linearly with the reduced frequency irrespectively of whether the airfoil is loaded or not. Simulations using a frequency domain linearized Navier-Stokes solver have been carried out on a low-pressure turbine airfoil section, the NACA0012 and NACA65 profiles and a flat plate operating at different flow conditions to show the generality and correctness of the analytical conclusions. Both the traveling-wave and influence coefficient formulations of the problem are used in combination to increase the understanding and explore the nature of the unsteady pressure perturbations.

Forced Response Analysis of High-Mode Vibrations for Mistuned Bladed Discs With Effective Reduced-Order Models

This paper is focused on the analysis of effects of mistuning on the forced response of gas-turbine bladed discs vibrating in the frequency ranges corresponding to higher modes. For high modes the blade aerofoils are deformed during vibrations and the blade mode shapes differ significantly from beam mode shapes. A model reduction technique is developed for the computationally efficient and accurate analysis of forced response for bladed discs vibrating in high frequency ranges. High-fidelity finite element models of a tuned bladed disc sector are used to provide primary information about dynamic properties of a bladed disc and the blade mistuning is modelled by specially defined mistuning matrices. The forced response displacement and stress amplitude levels are studied for high frequency ranges. The effects of different types of mistuning are examined and the existence of high amplifications of mistuned forced response levels is shown for high-mode vibrations: in some cases, the resonance peak response of a tuned structure can be lower than out-of-resonance amplitudes of its mistuned counterpart.