Forced Response Problem Research Articles

Stator Blades Manufacturing Geometrical Variability in Axial Compressors and Impact on the Aeroelastic Excitation Forces

AbstractThe manufacturing geometrical variability is a source of uncertainty, which cannot be avoided in the realization of machinery components. Deviations of a part geometry from its nominal design are inevitably present due to the manufacturing process. In the case of the aeroelastic forced response problem within axial compressors, these uncertainties may affect the vibration characteristics. For this reason, the impact of geometrical uncertainties due to the manufacturing process onto the modal forcing of axial compressor blades is investigated in this study. The research focuses on the vibrational behavior of an axial compressor rotor blisk. In particular, the amplitude of the forces acting as a source of excitation on the vibrating blades is studied. The geometrical variability of the upstream stator is investigated as input uncertainty. The variability is modeled starting from a series of optical surface scans. A stochastic model is created to represent the measured manufacturing geometrical deviations from the nominal model. A data reduction methodology is proposed in order to represent the uncertainty with a minimal set of variables. The manufacturing geometrical variability model allows to represent the input uncertainty and probabilistically evaluate its impact on the aeroelastic problem. An uncertainty quantification is performed in order to evaluate the resulting variability on the modal forcing acting on the vibrating rotor blades. Of particular interest is the possible rise of low engine orders due to the mistuned flow field along the annulus. A reconstruction algorithm allows the representation of the variability during one rotor revolution. The uncertainty on low harmonics of the modal rotor forcing can be therefore identified and quantified.

Validation of a Modal Work Approach for Forced Response Analysis of Bladed Disks

The paper describes a numerical method based on a modal work approach to evaluate the forced response of bladed disks and its validation against numerical results obtained by a commercial FEM code. Forcing functions caused by rotor–stator interactions are extracted from CFD unsteady solutions properly decomposed in time and space to separate the spinning perturbation acting on the bladed disk in a cyclic environment. The method was firstly applied on a dummy test case with cyclic symmetry where the forcing function distributions were arbitrarily selected: comparisons for resonance and out of resonance conditions revealed an excellent agreement between the two numerical methods. Finally, the validation was extended to a more realistic test case representative of a low-pressure turbine bladed rotor subjected to the wakes of two upstream rows: an IGV with low blade count and a stator row. The results show a good agreement and suggest computing the forced response problem on the finer CFD blade surface grid to achieve a better accuracy. The successful validation of the method, closely linked to the CFD environment, creates the opportunity to include the tool in an integrated multi-objective procedure able to account for aeromechanical aspects.

A novel approach to solve eigenvalue and forced response problems in waveguide theory by means of bi-orthogonality relations

In this paper recent advances on solving challenging problems in dynamics of multimodal symmetric waveguides (having identical properties of positive/negative going waves) using bi-orthogonality are summarised. It has already been shown by the authors that bi-orthogonality, valid for arbitrarily complicated unbounded symmetric waveguides, greatly facilitates analysis of their forced response in any excitation conditions. Recent findings have shown that bi-orthogonality is a powerful tool also for bounded waveguides and by direct use it resolves the classical Boundary Integral Equations (BIE) into a set of individual relations between modal amplitudes at the boundaries. Moreover, bi-orthogonality permits to uniquely identify special sets of boundary conditions for which the eigenfrequency spectra may be found directly from the dispersion diagrams. These solutions are available simply by drawing horizontal lines specifying an inverse of the wave length of interest in the (Ω, k)-dispersion diagram and so the eigenfrequency spectrum may be read directly as intersections of these lines with dispersion curves. This paper therefore serves to promote this method, discuss applications, new possibilities, physical interpretation and its generalisation to a broader class of waveguide problems.

Free and forced response of three-dimensional waveguides with rotationally symmetric cross-sections

The analysis of high-frequency wave propagation in waveguides of arbitrarily shaped cross-sections requires specific numerical methods. A rather common technique consists in discretizing the cross-section with finite elements while describing analytically the axis direction of the waveguide. This technique enables to account for the continuous translational invariance of a waveguide and leads to a modal problem written on the cross-section. Although two-dimensional, solving the so-obtained eigensystem can yet become computationally costly with the increase of the size of the problem. In most applications involving waveguides, the cross-section itself often obeys rules of symmetry, which could also be exploited in order to further reduce the size of the modal problem. A widely encountered type of symmetry is rotational symmetry. Typical examples are bars of polygon-shaped cross-section or multi-wire cables. The goal of this paper is to propose a numerical method that exploits the discrete rotational symmetry of the waveguide cross-section. Bloch–Floquet conditions are applied in the circumferential direction while the continuous translational invariance of the waveguide along its axis is still described analytically. Both the free and the forced response problems are considered. A biorthogonality relationship specific to the rotationally symmetric formulation is derived. Numerical results are computed and validated for the simple example of a cylinder and for the more complex test case of a multi-wire structure. In addition to reducing the computational effort, the rotationally symmetric formulation naturally provides a classification of modes in terms of their circumferential order, which can be of great help for the dynamic analysis of complex structures.

A modal approach based on perfectly matched layers for the forced response of elastic open waveguides

This paper investigates the computation of the forced response of elastic open waveguides with a numerical modal approach based on perfectly matched layers (PML). With a PML of infinite thickness, the solution can theoretically be expanded as a discrete sum of trapped modes, a discrete sum of leaky modes and a continuous sum of radiation modes related to the PML branch cuts. Yet with numerical methods (e.g. finite elements), the waveguide cross-section is discretized and the PML must be truncated to a finite thickness. This truncation transforms the continuous sum into a discrete set of PML modes. To guarantee the uniqueness of the numerical solution of the forced response problem, an orthogonality relationship is proposed. This relationship is applicable to any type of modes (trapped, leaky and PML modes) and hence allows the numerical solution to be expanded on a discrete sum in a convenient manner. This also leads to an expression for the modal excitability valid for leaky modes. The physical relevance of each type of mode for the solution is clarified through two numerical test cases, a homogeneous medium and a circular bar waveguide example, excited by a point source. The former is favourably compared to a transient analytical solution, showing that PML modes reassemble the bulk wave contribution in a homogeneous medium. The latter shows that the PML mode contribution yields the long-term diffraction phenomenon whereas the leaky mode contribution prevails closer to the source. The leaky mode contribution is shown to remain accurate even with a relatively small PML thickness, hence reducing the computational cost. This is of particular interest for solving three-dimensional waveguide problems, involving two-dimensional cross-sections of arbitrary shapes. Such a problem is handled in a third numerical example by considering a buried square bar.

Time-Linearized and Time-Accurate 3D RANS Methods for Aeroelastic Analysis in Turbomachinery

A computational method for performing aeroelastic analysis using either a time-linearized or an unsteady time-accurate solver for the compressible Reynolds averaged Navier--Stokes (RANS) equations is described. The time-linearized solver employs the assumption of small time-harmonic perturbations and is implemented via finite differences of the nonlinear flux routines of the time-accurate solver. The resulting linear system is solved using a parallelized generalized minimal residual (GMRES) method with block-local preconditioning. The time accurate solver uses a dual time stepping algorithm for the solution of the unsteady RANS equations on a periodically moving computational grid. For either solver, and both flutter and forced response problems, a mapping algorithm has been developed to map structural eigenmodes, obtained from finite element structural analysis, from the surface mesh of the finite element structural solver to the surface mesh of the finite volume flow solver. Using the surface displacement data an elliptic mesh deformation algorithm, based on linear elasticity theory, is then used to compute the grid deformation vector field. The developed methods are validated first using standard configuration 10. Finally, for an ultra-high bypass ratio fan, the results of the time-linearized and the unsteady module are compared. The gain in prediction time using the linearized methods is highlighted.

Bilinear modal representations for reduced-order modeling of localized piecewise-linear oscillators

A novel reduced-order modeling method for vibration problems of elastic structures with localized piecewise-linearity is proposed. The focus is placed upon solving nonlinear forced response problems of elastic media with contact nonlinearity, such as cracked structures and delaminated plates. The modeling framework is based on observations of the proper orthogonal modes computed from nonlinear forced responses and their approximation by a truncated set of linear normal modes with special boundary conditions. First, it is shown that a set of proper orthogonal modes can form a good basis for constructing a reduced-order model that can well capture the nonlinear normal modes. Next, it is shown that the subspace spanned by the set of dominant proper orthogonal modes can be well approximated by a slightly larger set of linear normal modes with special boundary conditions. These linear modes are referred to as bi-linear modes, and are selected by an elaborate methodology which utilizes certain similarities between the bi-linear modes and approximations for the dominant proper orthogonal modes. These approximations are obtained using interpolated proper orthogonal modes of smaller dimensional models. The proposed method is compared with traditional reduced-order modeling methods such as component mode synthesis, and its advantages are discussed. Forced response analyses of cracked structures and delaminated plates are provided for demonstrating the accuracy and efficiency of the proposed methodology.

Convergence predictions for aeroelastic calculations of tuned and mistuned bladed disks

Mistuning changes the dynamics of bladed disks significantly. Frequency domain methods for predicting the dynamics of mistuned bladed disks are typically based on iterative aeroelastic calculations. Converged aerodynamic stiffness matrices are required for accurate aeroelastic results of eigenvalue and forced response problems. The tremendous computation time needed for each aerodynamic iteration would greatly benefit from a fast method of predicting the number of iterations needed for converged results. A new hybrid technique is proposed to predict the convergence history based on several critical ratios and by approximating as linear the relation between the aerodynamic force and the complex frequencies (eigenvalues) of the system. The new technique is hybrid in that it uses a combined theoretical and stochastic/computational approach. The dynamics of an industrial bladed disk is investigated, and the predicted convergence histories are shown to match the actual results very well. Monte Carlo simulations using the new hybrid technique show that the aerodynamic ratio and the aerodynamic gradient ratio are the two most important factors affecting the convergence history.

Repetitive impact response of a beam structure subjected to harmonic base excitation

This paper investigates the forced response dynamics of a clamped–clamped beam to which a rigid body is attached, and in the presence of periodic or non-periodic impacts between the body and a comparatively compliant base structure. The assembly is subjected to base excitation at specified frequency and acceleration, and the potentially complex responses that occur are examined analytically. The two sets of natural frequencies and vibration modes of the beam-rigid body structure (in its in-contact state, and in its not-in-contact state), are used to treat the forced response problem through a series of algebraic mappings among those states. A modal analysis based on extended operators for the (continuous) beam and (discrete) rigid body establishes a piecewise linear state-to-state mapping for transition between the in-contact and not-in-contact conditions. The contact force, impulse, and displacement each exhibit complex response characteristics as a function of the excitation frequency. Periodic responses occurring at the excitation frequency, period-doubling bifurcations, grazing impacts, sub-harmonic regions, fractional harmonic resonances, and apparently chaotic responses each occur at various combinations of damping, excitation frequency, and contact stiffness. The results of parameter studies in structural asymmetry and the eccentricity of the contact point's location are discussed.

Nonlinear Analysis of Unsteady Flows in Multistage Turbomachines Using Harmonic Balance

A harmonic balance technique for the analysis of two-dimensional linear (small-disturbance) and nonlinear unsteady flows in multistage turbomachines is presented. The present method uses a mixed time-domain/frequency-domain approach that allows one to compute the unsteady aerodynamic response of multistage machines to both blade vibration (the flutter problem) and wake interaction (the forced response problem). In general, the flowfield may have multiple excitation frequencies that are not integer multiples of each other, so that the unsteady flow is (sometimes) aperiodic in time. Using our approach, we model each blade row using a computational grid spanning a single blade passage. In each blade row, we store several subtime level solutions. For flows that are periodic in time, these subtime levels span a single time period. For aperiodic flows, the temporal period spanned by these subtime level solutions is sufficiently long to sample the relevant discrete frequencies contained in the aperiodic flow. In both cases, these subtime level solutions are related to each other through the time-derivative terms in the Euler or Navier-Stokes equations and boundary conditions; complex periodicity conditions connect the subtime levels within a blade passage, and interrow boundary conditions connect the solutions among blade rows. The resulting discretized equations, which are mathematically steady because time derivatives have been replaced by a pseudospectral operator in which the excitation frequencies appear as parameters, can be solved very efficiently using multigrid acceleration techniques. In this paper, we apply the technique to both flutter and wake-interaction problems and illustrate the influence of neighboring blade rows on the unsteady aerodynamic response of a blade row.

Combined/simultaneous gust and oscillating turbine row unsteady aerodynamics in subsonic flow

This paper presents, for the first time, a detailed experimental comparison between combined and simultaneous motion- and gust-induced unsteady aerodynamic response of a turbine cascade. Experiments are performed in a non-rotating annular test facility to investigate the local validity of the linearised treatment of the forced response problem. The gust-response unsteady aerodynamics are experimentally modelled using rotating gust generators. The torsion mode unsteady aerodynamics of a turbine cascade are investigated by controlling the motion of all 20 blades. Combined gust and oscillating blades unsteady aerodynamics are obtained by the linear superposition of the oscillating blade row and the gust-response unsteady aerodynamics measured separately. Simultaneous gust- and motion-induced unsteady aerodynamic responses are obtained by oscillating the blades in presence of the 13 or 22 per revolution forcing function. The linear superposition principle could be shown to be locally valid for the investigated flow and vibration conditions.

Transonic Rotor/Stator Interaction Using a 3-D Linearised Unsteady Viscous Flow Model

The paper first summarizes the forced response problem in turbomachinery and reviews various numerical methods for the simulation of unsteady flows. A particular technique, based on the linearisation of the unsteady Favre-averaged Navier-Stokes equations on three-dimensional mixed-element grids of tetrahedra, hexahedra and wedges, is described in some detail. The methodology was applied to a NGV/rotor interaction benchmark case for which detailed steady and unsteady flow measurements are available. The steady-state flow, calculated using a non-linear viscous representation, was described in detail with emphasis on features such as separation, horseshoe and passage vortices, tip leakage and shock structure since these are likely to influence the unsteady flow. The sources of unsteadiness on the rotor passage were evaluated from the steady-state solution at the NGV outlet. The disturbances were split into vortical, entropic and potential waves, the Fourier components of which were considered separately. The summation of the vortical and entropic waves was used as a rotor inlet boundary condition in order to assess the wake/rotor unsteady interaction. Similarly, potential waves were used to study the potential/rotor interaction. The results obtained from these two types of unsteady interactions were superimposed and compared with experimental data. Good qualitative and, in most cases, quantitative agreement was obtained, a finding which suggests that the unsteady flowfield generated by the relative blade motion can be considered to be a quasi-linear phenomenon for the particular HP turbine studied. Finally, the mechanisms of wake/rotor and potential/rotor interactions were studied in some detail and it was concluded that the former was strong in the crown of the blade while the latter was dominant in the leading edge region.

Solving for Mistuned Forced Response by Symmetry

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.

The Influence of Neighboring Blade Rows on the Unsteady Aerodynamic Response of Cascades

In this paper, we present an analysis of the unsteady aerodynamic response of cascades due to incident gusts (the forced response problem) or blade vibration (the flutter problem) when the cascade is part of a multistage fan, compressor, or turbine. Most current unsteady aerodynamic models assume the cascade to be isolated in an infinitely long duct. This assumption, however, neglects the potentially important influence of neighboring blade rows. We present an elegant and computationally efficient method to model these neighboring blade row effects. In the present method, we model the unsteady aerodynamic response due to so-called spinning modes (pressure and vorticity waves), with each mode corresponding to a different circumferential wave number and frequency. Then, for each mode, we compute the reflection and transmission coefficients for each blade row. These coefficients can be obtained from any of the currently available unsteady linearized aerodynamic models of isolated cascades. A set of linear equations is then constructed that couples together the various spinning modes, and the linear equations are solved via LU decomposition. Numerical results are presented for both the gust response and blade vibration problems. To validate our model, we compare our results to other analytical models, and to a multistage vortex lattice model. We show that the effect of neighboring blade rows on the aerodynamic damping of vibrating cascades is significant, but nevertheless can be modeled with a small number of modes.

Blade Row Interaction Effects on Flutter and Forced Response

A procedure to calculate intrablade row unsteady aerodynamic interactions is developed that relies upon results from isolated blade row unsteady aerodynamic analyses. Using influence coefficients that express the unsteady forces on one blade row due to the motion of another, an aeroelastic model is obtained that accounts for the coupling of the vibratory responses of multiple blade rows. The model is applied to two model config- urations, each consisting of three blade rows. The flutter analysis shows that interaction effects can be desta- bilizing, and the forced response analysis shows that interaction effects can result in a significant increase in the resonant response of a blade row. N the flutter analysis of a turbomachine blade row, the blade row is commonly assumed to be isolated—distur- bances created by the vibrating blades are free to propagate away from the blade row without being disturbed. Therefore, any reflections of these outgoing waves by other structural members or nonuniformiti es in the mean-flowfield are ne- glected. Although the forced response problem is typically concerned with blade row interaction, forced response anal- yses also generally neglect any reflections of outgoing waves. However, in an engine environment, structural elements such as neighboring blade rows or struts and nonuniformities in the mean-flowfield will reflect some of this wave energy back toward the vibrating blades, causing additional unsteady forces on them. Whether or not these reflected waves can signifi- cantly affect the aeroelastic stability or forced response of a blade row is a question of fundamental importance. Several investigations have focused on the unsteady aero- dynamic interaction between two rigid blade rows. Kaji and Okazaki1 investigated the interaction of two blade rows for the purpose of predicting rotor-stator interaction noise. They obtained a simultaneous solution to the unsteady lift distri- butions on both of the blade rows. Hanson2 modified Smith's3 isolated blade row unsteady aerodynamic analysis to predict rotor-stator interaction noise, essentially extending Kaji and Okazaki's work to include effects of frequency scattering and mean-flow turning by the blade rows. For a counter-rotati ng propfan in incompressibl e flow, Chen and Williams4 used a panel method to determine the unsteady loads on rigid pro- peller blades. From the point-of-view of the present investi- gation, all of these investigations are somewhat limited be- cause they did not delve into the aeroelastic problem and they were limited to two blade rows. One investigation that did consider the aeroelastic effects of interactions of a vibrating blade row and an adjacent struc- ture was that of Williams et al. 5 A three-dimensio nal linear- ized compressible panel method was used to calculate the unsteady aerodynamics of a ducted fan. The aeroelastic anal- ysis allowed flexibility of both the fan and the duct. The duct was found to have a destabilizing effect on the fan. A number of time-accurate solutions to the Euler and Na- vier-Stokes equations for rotor-stator interaction have been

An Analysis System for Blade Forced Response

A frequent cause of turbomachinery blade failure is excessive resonant response. The most common excitation source is the nonuniform flow field generated by inlet distortion, wakes and/or pressure disturbances from adjacent blade rows. The standard method for dealing with this problem is to avoid resonant conditions using a Campbell diagram. Unfortunately, it is impossible to avoid all resonant conditions. Therefore, judgments based on past experience are used to determine the acceptability of the blade design. A new analysis system has been developed to predict blade forced response. The system provides a design tool, over and above the standard Campbell diagram approach, for predicting potential forced response problems. The incoming excitation sources are modeled using a semi-empirical rotor wake/vortex model for wake excitation, measured data for inlet distortion, and a quasi-three-dimensional Euler code for pressure disturbances. Using these aerodynamic stimuli, and the blade’s natural frequencies and mode shapes from a finite element model, the unsteady aerodynamic modal forces and the aerodynamic damping are calculated. A modal response solution is then performed. This system has been applied to current engine designs. A recent investigation involved fan blade response due to inlet distortion. An aero mechanical test had been run with two different distortion screens. The resulting distortion entering the fan was measured. With this as input data, the predicted response agreed almost exactly with the measured response. In another application, the response of the LPT blades of a counterrotating supersonic turbine was determined. In this case the blades were excited by both a wake and a shock wave. The shock response was predicted to be three times larger than that of the wake. Thus, the system identified a new forcing function mechanism for supersonic turbines. This paper provides a basic description of the system, which includes: (1) models for the wake excitation, inlet distortion, and pressure disturbance; (2) a kernel function solution technique for unsteady aerodynamics; and (3) a modal aeroelastic solution using strip theory. Also, results of the two applications are presented.

On the forced vibration of simply supported rotating cylindrical shells

The free and forced vibrations of rotating simply supported cylindrical shells are investigated. The equations of motion, which take into account the effects of initial tensions, are derived from Hamilton’s principle. The effects of rotation on the natural frequencies are first reviewed. Next, a new approach is developed for solving the general forced vibration response problem of rotating cylindrical shells. The solution in its final form is characterized by generalized coordinates. The orthogonality of trigonometric functions allows one to separate the solution such that the generalized coordinates associated with different rotational modes form a set of second-order differential equations and can be solved for independently. The solution is further simplified for the cases where flexural vibration predominates. An inverted problem, where the cylinder is stationary and the load is rotating around it, is solved too for comparison. Finally, analytical solutions to the constant point load, and to the harmonic point load in the radial direction, are obtained using the developed method. The influence of rotation on the forced vibration is then illustrated.

A Numerical Method for Solving the Forced Baroclinic Coastal-Trapped Wave Problem of General Form

Abstract The determination of baroclinic coastal-trapped wave modes reduces to an eigenvalue problem quadratic in the eigenvalue if the long-wave assumption is not made. This problem can be expressed as a linear eigenvalue problem of expanded dimension. Consideration of the expanded problem permits direct determination of the set of functions orthogonal to the baroclinic eigenmodes sought. Using this set, the forced baroclinic response problem can be expressed as a series of first-order, ondinary differential equations, one for each forced eigenmode. The method presented represents two significant advances. First, the solution technique permits the calculation of complex wavenumbers directly as they would result from the determination of evanescent solutions or of solutions including bottom friction. Second, the forcing of waves that are solutions to equations nonlinear in the eigenvalue can be determined. Although the method has been described in the context of baroclinic coastal-trapped waves, it has mo...

Flutter and forced response of mistuned rotors using standing wave analysis

A standing wave approach is applied to the analysis of the flutter and forced response of tuned and mistuned rotors. The traditional traveling wave cascade airforces are recast into standing wave arbitrary motion form using Pade approximants, and the resulting equations of motion are written in the matrix form. Applications for vibration modes, flutter, and forced response are discussed. It is noted that the standing wave methods may prove to be more versatile for dealing with certain applications, such as coupling flutter with forced response and dynamic shaft problems, transient impulses on the rotor, low-order engine excitation, bearing motions, and mistuning effects in rotors.