Forced Response Levels Research Articles

Nonlinear Vibration Analysis of Turbine Bladed Disks With Midspan Dampers

Abstract Friction dampers are one of the most common secondary structures utilized to alleviate excessive vibration amplitudes in turbomachinery applications. In this paper, the dynamic behavior of the turbine bladed disks coupled with one of the special damper designs, the so-called Midspan Dampers (MSDs) that is commonly used in steam turbines of Baker Hughes Company, is thoroughly studied. Friction between the blade and the damper is modeled through a large number of contact nodes by using two-dimensional contact elements with a variable normal load. In the solution procedure, the coupled static/dynamic Harmonic Balance approach is utilized for the first time in the assessment of the dissipation capability of MSDs, computationally shown by predicting the forced response levels of the system at different resonances. Moreover, it is demonstrated that the nonlinear dynamic response is non-unique and it may vary considerably even if all the user-controlled inputs are kept identical. This phenomenon is a novel observation for MSDs, and it is explained by an uncertainty present in the contact forces. Contact conditions corresponding to multiple responses are also investigated to unveil the different kinematics of the damper under the same nominal conditions.

Experimental and Numerical Assessment of Mistuning Effects on Vibratory Response of a Bladed Disk With Underplatform Dampers

Abstract This paper presents experimental and numerical investigation of mistuned forced responses of an integrally bladed disk with full set of underplatform dampers (UPDs). This research aims at providing: 1. An experimental benchmark for nonlinear dynamics of a mistuned bladed disks with UPDs. 2. A numerical model that can account for features of a mistuned forced response level. Accordingly, a detailed experimental campaign is conducted on a static test rig called Octopus. This rig is specifically designed to investigate the dynamics of a full-scale integrally bladed disk (blisk) with UPDs in a noncontact manner so that the dynamic response of the system is not modified. The effect of mistuning on experimental forced response levels is assessed and a linearized model is proposed to predict the modulation of frequency response functions (FRFs) due to the frequency splitting. In the development of the model, the mistuning pattern identified from the linear blisk without UPDs is used and it is assumed that adding the dampers does not change the structural mistuning of the blisk. In this study, the fundamental mistuning model identification (FMM ID) was employed to identify the mistuning pattern of the blisk. It is shown that the proposed model successfully predicts the modulation of linear mistuned FRFs. The linearized model is also able to predict the modulation of nonlinear mistuned FRFs in stick condition (when nonlinear friction damping is negligible) with a good accuracy validating this assumption that adding the dampers does not change the mistuning pattern.

Geometric Mistuning Reduced-Order Model Development Utilizing Bayesian Surrogate Models for Component Mode Calculations

AbstractIntegrally bladed rotors (IBRs) are meant to be rotationally periodic structures. However, unique variations in geometries and material properties from sector-to-sector, called mistuning, destroy the structural periodicity. This results in mode localization that can induce forced response levels greater than what is predicted with a tuned analysis. Furthermore, mistuning and mode localization are random processes that require stochastic treatments when analyzing the distribution of fleet responses. Generating this distribution can be computationally intensive when using the full finite element model (FEM). To overcome this expense, reduced-order models (ROMs) have been developed to accommodate fast calculations of mistuned forced response levels for a fleet of random IBRs. Usually, ROMs can be classified by two main families: frequency-based and geometry-based methods. Frequency-based ROMs assume mode shapes do not change due to mistuning. However, this assumption has been shown to cause errors that propagate to the fleet distribution. To circumvent these errors, geometry-based ROMs have been developed to provide accurate predictions. However, these methods require recalculating modal data during ROM formulations. This increases the computational expense in computing fleet distributions. A new geometry-based ROM is presented to reduce this cost. The developed ROM utilizes a Bayesian surrogate model in place of sector modal calculations required in ROM formulations. The method, surrogate modal analysis for geometry mistuning assessments (SMAGMA), will propagate uncertainties of the surrogate prediction to forced response. ROM accuracies are compared to the true forced response levels and results computed by a frequency-based ROM.

Evaluation of free interface-based reduction techniques for nonlinear forced response analysis of shrouded blades

The efficient dynamic stress assessment of turbine blades is of prime importance in turbomachinery design. An accurate prediction of forced response level of shrouded blades requires a very detailed finite element model in addition to a nonlinear solver. In order to perform nonlinear forced response analysis of blades at an affordable computational cost, applying a model order reduction technique is essential. The appeal for component mode synthesis methods in dimension reduction of structures with friction contacts is due to the possibility of retaining a subset of physical degrees of freedom (e.g. the contact degrees of freedom) in the set of generalized coordinates. In this paper, a reduction method recently developed for nonlinear forced response analysis of structures with local nonlinearity is evaluated and compared with two classical component mode synthesis reduction techniques. All three methods have the same projection basis, which includes residual flexibility attachment modes and free interface modes, but different implementation. The response is computed in the frequency domain using multiharmonic balance method and periodic contact forces are modeled with a node-to-node 3D friction contact model. In order to demonstrate the efficiency of the three formulations, a rod and a simplified shrouded turbine blade are considered as case studies.

Mixed-Boundary Component Mode Substitution for Nonlinear Dynamics of Mistuned Shrouded Bladed Disks

A new mixed-boundary model reduction technique is presented for nonlinear forced response analysis of mistuned bladed disks with shroud friction contacts. In contrast to conventional fixed- or free-mode component mode synthesis (CMS) techniques, blades are reduced in a mixed-boundary fashion. Accordingly, blade component normal modes with a more realistic mixed boundary condition (fixed at blade-disk interface and free at shroud contacts) are incorporated into the reduction basis. Afterward a modal synthesis based on loaded interface modeshapes of the disk is performed to reduce the disk component and the previously retained blade-disk interface degrees of freedom (DOFs). This results in a compact reduced order model (ROM) in which retained physical DOFs contain only shroud contact nodes. During the reduction steps, sector-level frequency mistuning is efficiently introduced into the ROM. In addition, the reduction basis is invariant of the mistuning pattern, which makes the ROM suitable for statistical characterization of nonlinear response levels. The final ROM is computed using one single sector only and with minimal computational effort. Numerical simulations revealed an improved accuracy of the ROM based on the mixed-boundary model in predicting the nonlinear forced response levels of a mistuned bladed disk with shroud friction contacts.

Mistuning sensitivity and optimization for bladed disks using high-fidelity models

Abstract An effective method is developed for efficient calculations of the sensitivity of the maximum forced response levels for mistuned bladed disks with respect to blade frequency mistuning. The expressions for 1st and 2nd order sensitivity coefficients are derived in an analytical form which provides high accuracy and computational efficiency. Then, the optimization methods are used for searching the best and worst mistuning patterns of bladed disks. Two major types of the mistuning optimization problems are considered: (i) a continuous optimization problem when the blade mistuning can take any values from a prescribed range and (ii) a combinatorial optimization problem, when the set of mistuned blades is given and the optimization can be achieved by blade re-arrangement in a disk. For the first type of the optimization problem a set of sensitivity-based optimization algorithms is applied and for the second type a variant of a genetic algorithm is developed. The analysis of mistuning sensitivity coefficients and results of optimization searching are shown on an example of a realistic turbine bladed disk.

Analysis of Forced Response for Bladed Disks Mistuned by Material Anisotropy Orientation Scatters

A new method is developed for the forced response analysis of mistuned bladed disks manufactured from anisotropic materials and mistuned by different orientations of material anisotropy axes. The method uses (i) sector finite element (FE) models of anisotropic bladed disks and (ii) FE models of single blades and allows the calculation of displacements and stresses in a mistuned assembly. A high-fidelity reduction approach is proposed which ensures high-accuracy modeling by introducing an enhanced reduction basis. The reduction basis includes the modal properties of specially selected blades and bladed disks. The technique for the choice of the reduction basis has been developed, which provides the required accuracy while keeping the computation expense acceptable. An approach for effective modeling of anisotropy-mistuned bladed disk without a need to create a FE model for each mistuning pattern is developed. The approach is aimed at fast statistical analysis based on Monte Carlo simulations. All components of the methodology for anisotropy-mistuned bladed disks are demonstrated on the analysis of models of practical bladed disks. Effects of anisotropy mistuning on forced response levels are explored.

On the dual Craig–Bampton method for the forced response of structures with contact interfaces

Assembled structures are characterized by contact interfaces that introduce a local non-linearity and affect the dynamics of the assembly in terms of resonance frequencies and vibration levels. To assess the forced response levels of the assemblies during the design, nonlinear dynamic analyses are performed and, in order to reduce the computation time, spatial and temporal reductions of the governing equations must be used. A classical way to achieve temporal reduction is to implement the harmonic balance method to turn the time-domain differential governing equations into frequency-domain algebraic equations. Due to the local nature of contact interfaces, which usually involve a subset of degrees of freedom (dofs) of the structure, a common strategy to achieve spatial reduction is to use component mode synthesis (CMS), by retaining the contact dofs as master dofs. In this paper, a recent CMS approach, named dual Craig–Bampton method (Rixen in J Comput Appl Math, 2004. doi: 10.1016/j.cam.2003.12.014 ), is applied to the nonlinear forced response of structures with contact interfaces. The spectral orthogonality of the two subsets of mode shapes used as a projection basis is exploited to write a set of algebraic equations of the contact dofs in the frequency domain, with no need to compute the reduced matrices of the system. Different formulations of the governing equations are proposed for different configurations (i.e., outer contacts, inner contacts and structures with floating components), and two academic numerical test cases are used to demonstrate the method.

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

This paper is focused on the analysis of effects of mistuning on the forced response of gas turbine engine bladed disks vibrating in the frequency ranges corresponding to higher modes. For high modes considered here, 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 disks vibrating in high-frequency ranges. The high-fidelity finite element (FE) model of a tuned bladed disk sector is used to provide primary information about dynamic properties of a bladed disk, and the blade mistuning is modeled by specially defined mistuning matrices. The forced response displacement and stress amplitude levels are studied. 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.

Next Generation Traveling-Wave Excitation System for Integrally Bladed Rotors

AbstractAn n-harmonic traveling-wave excitation (nTWE) system that was developed for the U.S. air force research laboratory turbine engine fatigue facility (TEFF) is described. This traveling-wave system is designed to simulate engine order excitations on stationary integrally bladed rotors (IBRs) and dual flow-path IBRs (DFIBRs) for the purpose of measuring forced response amplification due to mistuning. This new system extends beyond traditional traveling-wave systems by its capability to generate a combination of n engine order (EO) harmonics for IBRs as well as different EO harmonics simultaneously to both inner- and outer-blade sets of DFIBRs. This system can excite both IBRs and DFIBRs of differing sizes and number of blades with electromagnetic excitation. Scanning vibrometry mounted above the test specimen is used to measure any localization due to mistuning. Sources of error in the nTWE system are discussed and experimental forced response levels are shown for a 22-blade IBR.

Vaned Diffuser Induced Impeller Blade Vibrations in a High-Speed Centrifugal Compressor

Blade failure in turbomachinery is frequently caused by an excessive resonant response. Forced response of the blades typically originates from unsteady fluid structure interactions. This paper presents the experimental and computational results of a research effort focusing on the blade forced response in a high-speed centrifugal compressor caused by the downstream vaned diffuser. The potential field from the downstream vaned diffuser acts as an unsteady impeller relative circumferentially nonuniform disturbance. In this work the effect of varying the radial gap between impeller exit and diffuser vane leading edges was examined. Dynamic strain gauges, which were installed on the blade surfaces, were used to measure the forced response levels of the blades and to estimate the damping properties for different compressor operating conditions and vaneless gap dimensions. Unsteady fluid flow simulations were used to quantify the forcing function acting on the compressor blades due to impeller-diffuser interaction. The time-resolved blade pressure distribution showed the temporal evolution of the dynamic load on the blade surface caused by the diffuser's potential field. The magnitude of the vibratory stress levels was found to depend on the radial gap size, the blade damping properties, and on the compressor operating point. The variation of the radial gap size resulted in a shift of the impeller-diffuser interaction zone towards the main blade leading edge by up to 5% of the streamwise location.

Reduction of Forced Response Levels for Bladed Disks by Mistuning: Overview of the Phenomenon

The newly revealed phenomenon of reduction of forced response levels in a mistuned bladed disk to levels significantly (e.g., by a factor of 2 and more) lower than that of its tuned counterpart is studied in detail on an example of a realistic bladed disk. Statistical properties of the amplification factor of the mistuned forced response calculated with aero-effects included have been studied for cases of random blade mistuning and for mistuned blade rearrangements. The optimization search for the best mistuning patterns providing maximum forced response reduction effect have been performed and the robustness of the optimum mistuning patterns has been demonstrated. The combined effect of the aerodynamic and structural damping on the response reduction is assessed. It is shown that the new phenomenon is of major practical significance and has to be taken into account in analysis of the forced response and design decisions.

Mistuning Forced Response Characteristics Analysis of Mistuned Bladed Disks

The problem of determining the worst-case mistuning pattern and robust maximum mistuning forced response of a mistuned bladed rotor is formulated and solved as an optimization problem. This approach is exemplified on a two-degrees-of-freedom per blade disk model, two three-degrees-of-freedom per blade disk models, and a mistuned two-stage bladed rotor. The results of the optimum search of the worst-case mistuning patterns for the lumped parameter models are analyzed, which reveals that the maximum blade forced response in a mistuned bladed disk is associated with mistuning jump, which causes strong localization of the vibration response in a particular blade. The mistuning jump-localization phenomenon has been observed for all of the numerical examples, and it is also demonstrated that the highest response was always experienced by a blade of mistuning value jump. The two- and three-degrees-of-freedom per blade disk models are also for determination of its sensitivity coefficients with respect to mistuning variation. Studies show that there is not a threshold of mistuning beyond which the maximum forced response levels off, or even drops, as the degree of mistuning is increased further. The maximum magnification factor is found to increase as the mistuning level is increased and reaches a maximum value at the upper limit of the mistuning level. The influence of the multistage coupling is revealed by comparing the results of single-stage analysis with that of the multistage case. The computed results have been compared with the Monte Carlo simulation produced, and it is demonstrated that the accuracy and efficiency of the maximum amplitude magnification factor computed by the presented method can be better than that of Monte Carlo simulations.

Management of the variability of vibration response levels in mistuned bladed discs using robust design concepts. Part 1: Parameter design

Small unavoidable differences (e.g. 5%) between blades on a bladed disc, called mistuning, can lead to a huge variation of forced vibration response levels, and some of them are extremely high (e.g. 500% of the level experienced on every blade is all blades are identical). In this first half of a two-part article, a novel approach of designing a bladed disc with a lower chance of encountering high vibration response levels is evaluated. A robust design concept is applied to manage the variability of the vibration response levels, and the new approach resembles parameter design in Taguchi method of robust design. A “robustness map” is created using simulations results of a 6-degree-of-freedom (6-DOF) system, and such a map is validated by two more complicated models. The robustness map is used to explain the behaviour of bladed discs investigated in previous studies and to give possible methods of delivering more robust bladed disc designs.

Management of the variability of vibration response levels in mistuned bladed discs using robust design concepts. Part 2: Tolerance design

The mistuning pattern on a bladed disc is controlled in Part 2 of the two-part article either by (i) imposing a small maximum allowable mistune according to the small mistuning approach or (ii) incorporating non-identical blades of specific patterns, known as the intentional mistuning approach. These approaches resemble the tolerance design stage of the Taguchi method of robust design. The first-order maximum amplification factor sensitivity in a single-degree-of-freedom (DOF)-per-sector system is derived to support a new definition of the interblade coupling ratio and to illustrate the dependence of the maximum amplification factor sensitivity on design parameters of a bladed disc. It is found that the variability of the forced vibration response levels in flexible bladed discs can be reduced by controlling the degree of mistune within realistic levels. The potential of a “linear” mistuning pattern to become an effective intentional mistuning pattern is evaluated by observing the amplification factors of bladed discs with combined intentional mistuning and additional random mistuning. A tool based on the importance sampling method is used to reduce the computational effort in determining the magnitude of intentional mistuning. Guidelines of designing bladed discs with a lower variability of forced vibration response levels are given according to the findings in casting the blade mistuning problem as a robust design problem.

A Method for Forced Response Analysis of Mistuned Bladed Disks With Aerodynamic Effects Included

A method has been developed for high-accuracy analysis of forced response levels for mistuned bladed disks vibrating in gas flow. Aerodynamic damping, the interaction of vibrating blades through gas flow, and the effects of structural and aerodynamic mistuning are included in the bladed disk model. The method is applicable to cases of high mechanical coupling of blade vibration through a flexible disk and, possibly shrouds, to cases with stiff disks and low mechanical coupling. The interaction of different families of bladed disk modes is included in the analysis providing the capability of analyzing bladed disks with pronounced frequency veering effects. The method allows the use of industrial-size sector models of bladed disks for analysis of forced response of a mistuned structure. The frequency response function matrix of a structurally mistuned bladed disk is derived with aerodynamic forces included. A new phenomenon of reducing bladed disk forced response by mistuning to levels that are several times lower than those of their tuned counterparts is revealed and explained.

Forced Response of Mistuned Bladed Disks in Gas Flow: A Comparative Study of Predictions and Full-Scale Experimental Results

An integrated experimental-numerical study of forced response for a mistuned bladed disk has been performed. A full chain for the predictive forced response analysis has been developed including data exchange between the computational fluid dynamics code and a code for the prediction of the nonlinear forced response for a bladed disk. The experimental measurements are performed at a full-scale single stage test rig with excitation by aerodynamic forces from gas flow. The numerical modeling approaches and the test rig setup are discussed. Comparison of experimentally measured and predicted values of blade resonance frequencies and response levels for a mistuned bladed disk without dampers is performed. A good correspondence between frequencies at which individual blades have maximum response levels is achieved. The effects of structural damping and underplatform damper parameters on amplitudes and resonance frequencies of the bladed disk are explored. It is shown that the underplatform damper significantly reduces scatters in values of the individual blade frequencies and maximum forced response levels.

Underplatform dampers for turbine blades: The effect of damper static balance on the blade dynamics

In this paper, a novel method to compute the static balance of turbine blades with underplatform dampers is presented, focusing on the influence of the static loads on the system dynamics. The proposed method allows preventing the calculation of non-unique forced response levels of the blade, occurring when simplified hypotheses are applied to the damper static balance.

Method for Sensitivity Analysis of Resonance Forced Response of Bladed Disks With Nonlinear Contact Interfaces

An effective method has been developed to calculate the sensitivity of the resonance peak frequency and forced response level to variation of parameters of nonlinear friction contact interfaces and excitation. The method allows determination of the sensitivity characteristics simultaneously with the resonance peak frequency and response level calculated as a function of any parameter of interest and without significant computational expense. Capabilities of the method are demonstrated on examples of analysis of large-scale finite element models of realistic bladed disks with major types of the nonlinear contact interfaces: (i) a blisk with underplatform dampers, (ii) a bladed disk with friction damping at blade fir-tree roots, and (iii) a high-pressure bladed disk with shroud contacts. The numerical investigations show high efficiency of the method proposed.

Analysis of sensitivity and robustness of forced response for nonlinear dynamic structures

An effective method is developed for calculation of the sensitivity and robustness of the forced response levels for strongly nonlinear structures. The sensitivity coefficients are determined with respect to parameters of friction contact interfaces, parameters of linear components of an assembled structure, frequency and level of excitation forces. Equations for determination of first- and second-order sensitivity coefficients of the forced response are derived analytically from a nonlinear multiharmonic equation of motion. The analytical derivation allows accurate and fast evaluation of the sensitivity coefficients. The sensitivity coefficients are calculated in frequency domain for each excitation frequency over the frequency range analysed simultaneously with the force response levels. The developed highly efficient method allows calculation of sensitivity characteristics without a noticeable increase of the computation time in addition to the time required for the forced response calculation. A measure of the forced response robustness is introduced. Sensitivity-based method for assessment of the forced response robustness for given ranges of uncertainty of structural and operating parameters is proposed. The methodology developed is illustrated on a set of problems including cases of forced response analysis for realistic strongly nonlinear gas-turbine structures.