Novel parametric reduced order model for aeroengine blade dynamics

https://doi.org/10.1016/j.ymssp.2015.02.015Get rights and content

Highlights

  • Novel reduced order model (ROM) for mistuned blades.

  • ROM based on beam framework with parameters from balde dynamics.

  • Improves accuracy, sensitivity and CPU time costs of the modal prediction of blades.

Abstract

The work introduces a novel reduced order model (ROM) technique to describe the dynamic behavior of turbofan aeroengine blades. We introduce an equivalent 3D frame model to describe the coupled flexural/torsional mode shapes, with their relevant natural frequencies and associated modal masses. The frame configurations are identified through a structural identification approach based on a simulated annealing algorithm with stochastic tunneling. The cost functions are constituted by linear combinations of relative errors associated to the resonance frequencies, the individual modal assurance criteria (MAC), and on either overall static or modal masses. When static masses are considered the optimized 3D frame can represent the blade dynamic behavior with an 8% error on the MAC, a 1% error on the associated modal frequencies and a 1% error on the overall static mass. When using modal masses in the cost function the performance of the ROM is similar, but the overall error increases to 7%. The approach proposed in this paper is considerably more accurate than state-of-the-art blade ROMs based on traditional Timoshenko beams, and provides excellent accuracy at reduced computational time when compared against high fidelity FE models. A sensitivity analysis shows that the proposed model can adequately predict the global trends of the variations of the natural frequencies when lumped masses are used for mistuning analysis. The proposed ROM also follows extremely closely the sensitivity of the high fidelity finite element models when the material parameters are used in the sensitivity.

Introduction

A bladed disk typically consists of a set of disk/blade sectors that are designed to be identical. However, there are always small variations in the structural properties of individual blades due to manufacturing and assembly tolerances, material imperfections and operational wear and tear [1]. These dimensional and material uncertainties lead to variations of the blade natural frequencies from their nominal design value. This phenomenon is generally denoted as blade mistuning. Mistuning in the free response splits the repeated natural frequencies associated with circumferential modes and distorts the corresponding mode shapes [2], [3], [4], [5], [6]. Simultaneously, these circumferential mode shapes increase the harmonic content of nodal diameters, leading to coupling with engine-induced vibrations [7]. In the worst case scenario mistuning also causes mode localization phenomena, in which the vibrational energy is transferred and confined to a subset of blades. This phenomenon may result in dynamic deformations significantly larger than those estimated at the design stage [8], [9]. As a consequence, mistuning compromises the high-cycle fatigue endurance of bladed disks and reduces the durability and the reliability of the entire engine. As a result, the research on the attenuation of mistuning has been a key activity in aero-engine R&D for more than 50 years, although the mistuning problem is considered still unsolved [2].

High-fidelity finite element models of bladed disks are used to predict both the maximum dynamic excitation at representative design points and the associated behavior due to pre-defined mistuning patterns [10], [11]. From a general perspective, analytical and numerical models of mistuning are more cost-effective than direct experimental characterizations because a large number of tests on nominally identical bladed disks are necessary to have a statistically representative population for all the various sources of uncertainty leading to mistuning. Since mistuning breaks the cyclic symmetry of bladed-disk systems [7], simplified FE models of single sectors are not suitable for analysis purposes and in most cases a full bladed disk model is necessary. A complete FE model of a bladed disk however typically involves the use of millions of degrees of freedom (DOFs), making parametric analyses too expensive even on state-of-the-art high-performance computing facilities. The potential use of probabilistic approaches (i.e. Monte Carlo simulations, even by using improved techniques of sampling generation such as Latin hypercube) further increases the computational costs. As a consequence, various FE-based reduced order methods (ROM) have been developed in the last two decades.

ROMs used in mistuning applications can be broadly classified into two groups, which are both based on modal reduction techniques. The first group developed between 1983 and 2000 consists essentially in Component Mode Synthesis (CMS) techniques [12], [13], [14]. CMS approaches assume that the blades and the disk are distinct sub-structural elements, thus their modes are separately calculated via deterministic FE analyses. The modal bases of the two substructures are then employed to reduce the size of the overall system matrices by enforcing compatibility conditions at the interfaces between the blades and the disk. The compatibility between the dynamic displacements existing within the frontiers of substructures can be described by means of fixed-interface [15], free-interface [12] and hybrid-interface methods [13]. The type of interface used affects the numerical convergence of the ROM in a significant way. Fixed interface can exhibit very good convergence properties by increasing the number of component modes, but it requires large numbers of interface DOFs and therefore penalizes the computational efficiency [16]. Another limitation of the fixed interface method is the inability to obtain easily the required constrained modes from testing, making this method used primarily in analytical or purely numerical models [17]. Free interface methods have a slow convergence, although in principle this problem can be overcome by using residual flexibility (attachment modes) provided by low frequency approximations to describe the contribution from neglected high frequency modes [16]. Unfortunately, it is difficult to extract accurate residual flexibility terms from test data, as well as information on residuals in analytical model [17]. The second group of approaches initially introduced by Yang and Griffin in 2001 [18] are denoted as System Mode based Methods (SMM). These include the Fundamental Mistuning Model [19], the Component Mode Mistuning [20] and the Integral Mode Mistuning [2]. The main concept behind SMM techniques is to employ selected sets of tuned system modes as a basis to represent the tuned disk–blade system, and then add a perturbation matrix to represent the mistuning effect. This strategy in principle allows an exact representation of the baseline tuned system, and SMM techniques are also more computationally efficient than CMS approaches. The accuracy of the SMM is very much dependent on the modal representation of the tuned system, and practice in the industry has revealed the use of SMM as reliable predictive tool for the first six mistuned modes, which involves the extraction of hundreds of modes from the FE full-scale models. CMS and SMM models are moreover not able to perform the mistuning analysis by direct perturbation of specific parameters in the modal domain, but they generally introduce the mistuning in the form of a natural frequency deviation, which complicates the sensitivity studies required for robust design. Most importantly in sensitivity analyses, all these ROMs have to be rebuilt or updated through a repeated extraction of modes from perturbed FE models to guarantee accurate predictions, thus reducing the computational efficiency. Current state-of-the-art parametric analyses for mistuning problems often employ lumped parameter approaches. Although these models can reproduce some of the basic dynamics characteristics of bladed disk systems and can be used for the statistical analysis of mistuning forced-response behavior, their practical applicability is limited at best to the first three modes [7]. In terms of classical reduction techniques used in the industry, the accuracy provided by Improved Reduced System (IRS) techniques largely depends on the selection of numbers and locations of the master degree of freedoms (MDOFs) [21], as in classical Guyan reduction techniques [22]. Moreover, they are not able to exploit the cyclic nature of the geometry of bladed disc systems.

A consistent body of literature has also been devoted to investigate the effects generated by uncertainties associated to the geometry or material properties on the dynamics of the blade alone [1]. The blade models used in these works are mainly based on high-fidelity FE models, or coordinate measurement machine data [1], [23], [24]. State-of-the-art in ROMs to represent single rotating blades mainly consists in using simple cantilever beams with rectangular cross section to approximate the exact shape by finite element representation [25], [26]. Torsion-induced displacement are present in the vibrational patterns of these blades, and they are mainly induced by the coupling between the bending deformations in the flap wise and chord wise directions. Second order effects (like shear deformations, rotary inertia, warping of the cross section, root fixing and Coriolis accelerations) do also contribute in providing a complex dynamic displacement pattern. Fan and compressor blades have been traditionally modeled using either simple twisted Euler–Bernoulli’s beam approaches, or twisted Timoshenko’s beam formulation with varying levels of complexity and warping [27], [28]. Rotating pre-twisted blade dynamics for the two types of structural beams has also been thoroughly investigated [28]. Although these simplified models are able to capture the first few fundamental modes (global bending, torsion, axial stretching and coupled bending motions in two directions), they are less suitable to represent the mode shapes of real blades, characterised by strongly coupled bending and torsional deformations, chord wise bending and edge wise flap. Designers in the field have also pointed out that beam models provide a poor performance when it is required to predict the blade modes associated with chord wise bending, i.e. second strip mode [27]. The uncertainty propagation analysis of these models is mainly restricted to the material properties and lumped mass at particular positions for the first few global modes.

This paper addresses the development of a parametric reduced order model of a blade using a novel approach consisting in introducing a simplified structural layout (frame structure) that provides a dynamic behavior equivalent to that of the full-scale blade. The ROM model developed in this paper is primarily meant to be employed in sensitivity analyses, where the uncertainties associated to either blades from material properties’ deviation and geometrical mistuning or the joints due to manufacturing, assembly tolerances and blade/disk loosening during the high speed rotation are considered [29]. As a case study, we consider a metallic fan blade described using a “high-fidelity” (HF) solid FE modelling that provides the baseline characterization of the first six global modes. The ROM frame model is developed through a structural identification approach based on a simulated annealing algorithm with stochastic tunneling, and also benchmarked against an equivalent Timoshenko beam ROM as a reference. The equivalent frame concept can include either 2D or 3D beam kinematics. An optimization process involving the geometric configuration of the frame (ROM) model is then introduced to minimize the natural frequency (NF), MAC and overall mass/modal mass errors with respect to the full blade FE model. The identification of the ROM parameters is carried out with two different cost functions. The first includes the total static mass of the blade, while the second consider the modal masses [30], [31]. While the objective that involves the total static mass reflects a physical equivalence between the ROM and the baseline blade model, the use of the modal masses is representative of the effective inertia associated to each mode. As it will be shown, the results of the ROMs optimised according to the two different sets of objectives do differ and the static mass based optimisation leads to better results than those associated with the modal masses. This indicates that the identification of the ROM parameters via optimization procedures should consider the total static mass as an objective function in order to minimise the errors associated to natural frequencies and Modal Assurance Criteria (MAC). Finally, a comparison of the sensitivity performance of the ROM beam frame and the high fidelity FE model with lumped masses and perturbation of the material parameters is carried out to validate the new ROM concept for mistuning applications, with uncertainty distributions associated to the blade.

Section snippets

Reference blade model

The HF reference FE model of the blade is built using the commercial software ANSYS Rel. 11.0 (Ansys Inc., 2008) (Fig. 1). The model consists in 540 SOLID95 8-node elements. Each node has three translational DOFs. The total number of DOFs is 32,400. The blade is entirely made of Titanium, considered as a homogenous and isotropic material. The blade has a low slenderness ratio, i.e. 4, and an increasing twist from root to tip.

The HF model is clamped at the root by zeroing all DOFs. A modal

Classical Timoshenko beam ROM approach

The Timoshenko beam approach is applied to the blade considered in this work because the length to width ratio is smaller than five, and it is therefore necessary to include the effects of shear deformation and rotatory inertial to the structure. A single and straight Timoshenko beam model with three elements is applied, with its relevant stiffness and mass matrices derived from standard open literature. The material properties are assumed as the ones of titanium (Young’s modulus of 1.15e5 MPa,

The frame-based approach

The ROM proposed in this paper is based on a frame consisting of 9 individual beams. The rationale for this choice is justified by the fact that the geometry of each individual beam can be tailored to represent complex mode shapes, e.g. those for which bending and twisting are coupled. Moreover, a frame retains a sufficient level of geometrical similarity with the original blade. This fact allows a direct mapping of the displacements at prescribed locations on the blade to points belonging to

Initial value determination

In order to accelerate the ROM calibration, it is necessary to identify a “suitable” initial solution. This implies finding an initial reasonable guess for the cross-sectional parameters featuring the ROM beams. This could be done, for example, by imposing the equivalence of the total elastic potential and kinetic energy between the ROMs and the full HF FE model [35]. However, this equivalence requires the use of specific numerical models as the source of these data, therefore it is not

Optimization process

The optimization process is carried out starting from the initial configuration of the frame identified with the IRS technique described above. The cost function consists of the linear combination of three objective functions:Costfunction=i=13wiObji

The first two objectives are the relative error on the NF and the MAC between the ROMs and the HF FE model:Obj1=i=16{(pi(ωiaωi)/ωi)2}Obj2=i=16(1MAC(i))2

The pi factors in Eq. (15) represent additional weights. Since the NF tends due increase with

Optimization with static mass constraint—Results and discussions

Fig. 7a and b show the Pareto fronts of the 2D and 3D ROMS extracted from 40,000 feasible solutions. The colored map expresses the values of the static mass errors. CPU times are also indicated. The optimal solutions cluster near the Pareto front. This shows the sensitivity to different combinations of NF and MAC errors, depending on the weight ratio imposed in the cost function. The two models provide similar performance for a total mass error below 1%. The location of the Pareto front in the

Optimization with modal mass constraints—Results and discussions

Table 3 shows the distribution of the modal masses extracted from the HF FE model, calculated using Eq. (12). It can be observed that the first six modes are associated to 88% of the total inertia of the blade. This proves that considering the first six modes provides a good approximation of the overall blade dynamics. With the increase of the number of modes the share of the inertia will not be significantly modified. and the penalty costs for the computation of the ROM would increase.

The

Modes comparison and sensitivity study

Table 4 shows the comparison of the natural frequencies and relative errors calculated through the Timoshenko beam ROM and the 3D beam frame against the HF FE model for the first six modes. The 3D beam ROM frame has a significantly lower relative error compared to the Timoshenko beam ROM model for the last first three modes. The direct comparison between mode shapes from the beam frame and the HF FE model is also shown in Fig. 12. The 3D beam frame ROM model is not only able to capture the

Conclusions

The main objective of this work is the development of a parametric ROM representing a typical aero-engine blade. The ROM consists in a simplified structural layout that provides an equivalent dynamic behavior of the full-scale blade for the first six modes. The ROM concept makes use of 2D and 3D Euler–Bernoulli frames, whose beam elements are connected at locations that correspond to the experimental measurement points of frequency response functions on a real blade. The ROM frame concept can

Acknowledgements

The authors would like to acknowledge the support of Rolls-Royce plc for the support of this research through the Composites University Technology Centre (UTC) at the University of Bristol, UK. The Authors also acknowledge the support from the Strategic Investment in Low carbon Engine Technology (SILOET) programme supported by Rolls-Royce plc & the Technology Strategy Board (TSB). JY is also grateful to the China Scholarship Council for the support. The authors also thank the Reviewers for

References (41)

  • R. Levin et al.

    Dynamic finite element model updating using simulated annealing and genetic algorithms

    Mech. Syst. Signal Process.

    (1998)
  • D.J. Ewins

    Control of vibration and resonance in aero engines and rotating machinery–An overview

    Int. J. Pressure Vessels Piping

    (2010)
  • A. Sinha et al.

    Vibratory parameters of blades from coordinate measurement machine data

    J. Turbomach.

    (2008)
  • C. Hodges et al.

    Vibration isolation from irregularity in a nearly periodic structure: theory and measurements

    J. Acoust. Soc. Am.

    (1983)
  • M.P. Castanier et al.

    Modeling and analysis of mistuned bladed disk vibration: current status and emerging directions

    J. Propul. Power

    (2006)
  • J. MacBain et al.

    Maximum resonant response of mistuned bladed disks

    J. Vibr. Acoust. Stress Reliab. Des.

    (1984)
  • D. Whitehead

    The maximum factor by which forced vibration of blades can increase due to mistuning

    J. Eng. Gas Turbines Power

    (1998)
  • M. Imregun, D. Ewins, Aeroelastic vibration analysis of tuned and mistuned bladed systems, in: Proceedings of the...
  • H. Irretier

    Spectral analysis of mistuned bladed disk assemblies by component mode synthesis

  • G. Óttarsson

    Dynamic Modeling and Vibration Analysis of Mistuned Bladed Disks

    (1994)
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