Maximum Amplification of Blade Response due to Mistuning: Localization and Mode Shape Aspects of the Worst Disks

2003 ◽  
Vol 125 (3) ◽  
pp. 442-454 ◽  
Author(s):  
Alejandro J. Rivas-Guerra ◽  
Marc P. Mignolet
Keyword(s):  
Mode Shape ◽  
Parametric Study ◽  
Resonant Mode ◽  
Forced Response ◽  
Mode Shapes ◽  
Optimization Strategy ◽  
Bladed Disks ◽  
Worst Case ◽  
Disk Model ◽  
Maximum Amplification

This paper focuses on the determination and study of the maximum amplification of the steady state forced response of bladed disks due to mistuning. First, an optimization strategy is proposed in which partially mistuned bladed disks are considered as physical approximations of the worst case disk and the mistuned properties are sought to maximize the response of a specific blade. This approach is exemplified on both a reduced order model of a blisk and a single-degree-of-freedom per blade disk model an extensive parametric study of which is conducted with respect to blade-to-blade coupling, damping, and engine order. A mode shape-based formulation of the amplification factor is then developed to clarify the findings of the parametric study in the strong coupling/small damping limit. In this process, the upper bound of Whitehead is recovered for all engine orders and number of blades and the conditions under which this limit is exactly achieved or closely approached are clarified. This process also uncovers a simple yet reliable approximation of the resonant mode shapes and natural frequencies of the worst disk.

Maximum Amplification of Blade Response Due to Mistuning: Localization and Mode Shapes Aspects of the Worst Disks

2002 ◽  
Author(s):  
Alejandro J. Rivas-Guerra ◽  
Marc P. Mignolet
Keyword(s):  
Parametric Study ◽  
Resonant Mode ◽  
Forced Response ◽  
Mode Shapes ◽  
Optimization Strategy ◽  
Bladed Disks ◽  
Worst Case ◽  
Disk Model ◽  
Engine Order ◽  
Maximum Amplification

This paper focuses on the determination and study of the maximum amplification of the steady state forced response of bladed disks due to mistuning. First, an optimization strategy is proposed in which partially mistuned bladed disks are considered as physical approximations of the worst case disk and the mistuned properties are sought to maximize the response of a specific blade. This approach is exemplified on both a reduced order model of a blisk and a single-degree-of-freedom per blade disk model an extensive parametric study of which is conducted with respect to blade-to-blade coupling, damping, and engine order. A mode shape-based formulation of the amplification factor is then developed to clarify the findings of the parametric study in the strong coupling/small damping limit. In this process, the upper bound of Whitehead is recovered for all engine orders and number of blades and the conditions under which this limit is exactly achieved or closely approached are clarified. This process also uncovers a simple yet reliable approximation of the resonant mode shapes and natural frequencies of the worst disk.

Maximum Amplification of Blade Response Due to Mistuning in Multi-Degree-of-Freedom Blade Models

2004 ◽  
Author(s):  
Bing Xiao ◽  
Alejandro J. Rivas-Guerra ◽  
Marc P. Mignolet
Keyword(s):  
Amplification Factor ◽  
Stress Component ◽  
Closed Form Solution ◽  
Forced Response ◽  
Optimization Techniques ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Disk Model ◽  
Overall Response ◽  
Maximum Amplification

This paper focuses on the determination and study of the maximum amplification of the steady state forced response of bladed disks due to mistuning. A general multi-degree-of-freedom dynamic model is adopted for each blade/disk sector and optimization techniques are used to maximize a weighted quadratic norm of the response of the degrees-of-freedom of blade 1 (overall response of blade 1). First, a mathematical optimization effort is conducted in which the resonant mistuned mode shape(s) (1 for engine orders 0 and N/2 where N is the number of blades, 2 otherwise) is selected to maximize the overall response of blade 1. The form of these optimum mode shapes is derived for all weighting matrices. The specific mode shapes are also derived for two particular weights the first one of which depends on the tuned bladed disk mass matrix and for which the overall response is akin to the kinetic energy. A closed form solution is also derived when the analysis focuses solely on the response of a specific degree-of-freedom or a specific stress component. In these cases, the ratio of the corresponding overall response to its tuned counterpart, i.e. the amplification factor, is found to be the product of two terms. The first one is an amplification obtained by tuned variations of the blade properties/mode shapes and thus is referred to as the modal amplification factor. The second term is an amplification obtained by proper mistuning. Interestingly, the modal amplification factor may take on very large values while a representative value of the largest mistuned factor is often the Whitehead limit of (1+N)/2 as in the single-degree-of-freedom per blade model. The above formulation and results are readily extended to the optimization of the blade alone response (as opposed to blade and disk sector). Numerical optimization efforts were also undertaken on both a two-degree-of-freedom per blade disk model and a 24-blade blisk reduced order model. The results of these computational efforts not only confirm the assumptions and findings of the theoretical developments but also demonstrate that substantially larger amplification factors can be obtained with a general natural frequency mistuning as opposed to Young’s modulus mistuning. Finally, an amplification due to mistuning (no tuned amplification) slightly larger than the Whitehead limit was obtained with relative variations in blade alone frequencies less than 0.5%.

A Method for Use of Cyclic Symmetry Properties in Analysis of Nonlinear Multiharmonic Vibrations of Bladed Disks

2004 ◽  
Vol 126 (1) ◽  
pp. 175-183 ◽  
Author(s):  
E. P. Petrov
Keyword(s):  
High Efficiency ◽  
Equations Of Motion ◽  
Linear Part ◽  
Forced Response ◽  
Mode Shapes ◽  
Solution Technique ◽  
Disk Model ◽  
Sector Model ◽  
Bladed Disk ◽  
Symmetry Properties

An effective method for analysis of periodic forced response of nonlinear cyclically symmetric structures has been developed. The method allows multiharmonic forced response to be calculated for a whole bladed disk using a periodic sector model without any loss of accuracy in calculations and modeling. A rigorous proof of the validity of the reduction of the whole nonlinear structure to a sector is provided. Types of bladed disk forcing for which the method may be applied are formulated. A multiharmonic formulation and a solution technique for equations of motion have been derived for two cases of description for a linear part of the bladed disk model: (i) using sector finite element matrices and (ii) using sector mode shapes and frequencies. Calculations validating the developed method and a numerical investigation of a realistic high-pressure turbine bladed disk with shrouds have demonstrated the high efficiency of the method.

Intentional Response Reduction by Harmonic Mistuning of Bladed Disks With Aerodynamic Damping

2018 ◽  
Author(s):  
Sebastian Willeke ◽  
Lukas Schwerdt ◽  
Lars Panning-von Scheidt ◽  
Jörg Wallaschek
Keyword(s):  
Wave Train ◽  
Wave Mode ◽  
Forced Response ◽  
Mode Shapes ◽  
Bladed Disks ◽  
Nodal Diameter ◽  
Aerodynamic Damping ◽  
Mode Pair ◽  
Localized Mode ◽  
Amplitude Level

A harmonic mistuning concept for bladed disks is analyzed in order to intentionally reduce the forced response of specific modes below their tuned amplitude level. By splitting a mode pair associated with a specific nodal diameter pattern, the lightly damped traveling wave mode of the nominally tuned blisk is superposed with its counter-rotating complement. Consequently, a standing wave is formed in which the former wave train benefits from an increase in aerodynamic damping. Unlike previous analyses of randomly perturbed configurations, the mode-specific stabilization is intentionally promoted through adjusting the harmonic content of the mistuning pattern. Through a re-orientation of the localized mode shapes in relation to the discrete blades, the response is additionally attenuated by an amount of up to 7.6 %. The achievable level of amplitude reduction is analytically predicted based on the properties of the tuned system. Furthermore, the required degree of mistuning for a sufficient separation of a mode pair is derived.

Blade Root Joint Modelling and Analysis of Effects of Their Geometry Variability on the Nonlinear Forced Response of Tuned and Mistuned Bladed Disks

2020 ◽  
Author(s):  
Adam Koscso ◽  
E. P. Petrov
Keyword(s):  
Harmonic Balance Method ◽  
Forced Response ◽  
High Fidelity ◽  
Contact Interactions ◽  
Bladed Disks ◽  
Element Mesh ◽  
Disk Model ◽  
Bladed Disk ◽  
Nonlinear Contact ◽  
Manufacturing Tolerances

Abstract One of the major sources of the damping of the forced vibration for bladed disk structures is the micro-slip motion at the contact interfaces of blade-disk joints. In this paper, the modeling strategies of nonlinear contact interactions at blade roots are examined using high-fidelity modelling of bladed disk assemblies and the nonlinear contact interactions at blade-disk contact patches. The analysis is performed in the frequency domain using multiharmonic harmonic balance method and analytically formulated node-to-node contact elements modelling frictional and gap nonlinear interactions. The effect of the number, location and distribution of nonlinear contact elements are analyzed using cyclically symmetric bladed disks. The possibility of using the number of the contact elements noticeably smaller than the total number of nodes in the finite element mesh created at the contact interface for the high-fidelity bladed disk model is demonstrated. The parameters for the modeling of the root damping are analysed for tuned and mistuned bladed disks. The geometric shapes of blade roots and corresponding slots in disks cannot be manufactured perfectly and there is inevitable root joint geometry variability within the manufacturing tolerances. Based on these tolerances, the extreme cases of the geometry variation are defined and the assessment of the possible effects of the root geometry variation on the nonlinear forced response are performed based on a set of these extreme cases.

Design Methods of Friction Damping at Blade-Disk Joints

2009 ◽  
Author(s):  
Jie Hong ◽  
Lulu Chen ◽  
Yanhong Ma ◽  
Xin Yang
Keyword(s):  
Low Frequency ◽  
Harmonic Balance Method ◽  
Large Error ◽  
Forced Response ◽  
Mode Shapes ◽  
Bladed Disks ◽  
Friction Dampers ◽  
Tracking Motion ◽  
Nonlinear Force ◽  
Free Mode

Friction at blade-disk joints is an important source of damping that reduces low frequency resonant amplitudes to acceptable levels in blade-disk assemblies. An effective method is proposed to predict nonlinear forced response of bladed disks taking account of the nonlinear force at blade-disk joints in frequency domain, which syncretizes the excellencies of harmonic balance method, dynamic softness method and tracking motion method. Constrained Mode Shapes are introduced to express the relative motion which occurs at the contact interfaces of blade roots. Compared to using free mode shapes, fewer number of constrained mode shapes is required in order to obtain the accurate resonant response of a system with friction dampers when the contact state is fully stick. It is more efficient to predict the nonlinear forced response of bladed disks taking account of the nonlinear force at blade-disk joints. Based on this method, the effect of Boundary Conditions on the resonant frequencies and forced response levels under different engine rotational speeds is investigated. Large error in the prediction of forced response levels under low engine rotational speed by using traditional methods is found. The effects of preload distribution at blade roots and excitation level are also investigated.

scholarly journals Identification of Mistuning Characteristics of Bladed Disks From Free Response Data

1998 ◽  
Author(s):  
Marc P. Mignolet ◽  
Alejandro Rivas-Guerra
Keyword(s):  
Dynamic Models ◽  
Natural Frequencies ◽  
Dynamic Properties ◽  
Structural Parameters ◽  
Identification Problem ◽  
Forced Response ◽  
Mode Shapes ◽  
Likelihood Principle ◽  
Bladed Disks ◽  
Block Test

The focus of the present investigation is on the estimation of the dynamic properties, i.e. masses, stiffnesses, natural frequencies, mode shapes and their statistical distributions, of turbomachine blades to be used in the accurate prediction of the forced response of mistuned bladed disks. As input to this process, it is assumed that the lowest natural frequencies of the blades alone have been experimentally measured, for example in a broach block test. Since the number of measurements is always less than the number of unknowns, this problem is indeterminate in nature. Two distinct approaches will be investigated to resolve the shortfall of data. The first one relies on the imposition of as many constraints as needed to insure a unique solution to this identification problem. Specifically, the mode shapes and modal masses of the blades are set to their design/tuned counterparts while the modal stiffnesses are varied from blade-to-blade to match the measured natural frequencies. The second approach, based on the maximum likelihood principle, yields estimates of all the structural parameters of the blades through the minimization of a specified “cost function”. The accuracy of these two techniques in predicting the forced response of mistuned bladed disks will be assessed on simple dynamic models of the blades.

Experimental Investigation of Mode Localization and Forced Response Amplitude Magnification for a Mistuned Bladed Disk

2000 ◽  
Author(s):  
John Judge ◽  
Christophe Pierre ◽  
Oral Mehmed
Keyword(s):  
Experimental Investigation ◽  
Traveling Wave ◽  
Forced Response ◽  
Response Amplitude ◽  
Mode Shapes ◽  
Bladed Disks ◽  
Element Analysis ◽  
Mode Localization ◽  
Bladed Disk ◽  
Out Of Plane

The results of an experimental investigation on the effects of random blade mistuning on the forced dynamic response of bladed disks are reported. The primary aim of the experiment is to gain understanding of the phenomena of mode localization and forced response blade amplitude magnification in bladed disks. A stationary, nominally periodic, twelve-bladed disk with simple geometry is subjected to a traveling-wave, out-of-plane, “engine order” excitation delivered via phase-shifted control signals sent to piezo-electric actuators mounted on the blades. The bladed disk is then mistuned by the addition of small, unequal weights to the blade tips, and it is again subjected to a traveling wave excitation. The experimental data is used to verify analytical predictions about the occurrence of localized mode shapes, increases in forced response amplitude, and changes in resonant frequency due to the presence of mistuning. Very good agreement between experimental measurements and finite element analysis is obtained. The out-of-plane response is compared and contrasted with the previously reported in-plane mode localization behavior of the same test specimen. This work also represents an important extension of previous experimental study by investigating a frequency regime in which modal density is lower but disk-blade interaction is significantly greater.

scholarly journals An Experimental Investigation of Vibration Localization in Bladed Disks: Part II — Forced Response

1997 ◽  
Author(s):  
Marlin J. Kruse ◽  
Christophe Pierre
Keyword(s):  
Experimental Investigation ◽  
Forced Response ◽  
Mode Shapes ◽  
Bladed Disks ◽  
Structural Coupling ◽  
Vibration Localization ◽  
Bladed Disk ◽  
Engine Order ◽  
Localized Mode ◽  
The Impact

The results of an experimental investigation on the effects of random blade mistuning on the forced dynamic response of bladed disks are reported. Two experimental specimens are considered: a nominally periodic twelve-bladed disk with equal blade lengths, and the corresponding mistuned bladed disk, which features slightly different blades of random lengths. Both specimens are subject to traveling-wave excitations delivered by piezo-electric actuators. The primary aim of the experiment is to demonstrate the occurrence of an increase in forced response blade amplitudes due to mistuning, and to verify analytical predictions about the magnitude of these increases. In particular, the impact of localized mode shapes, engine order excitation, and disk structural coupling on the sensitivity of forced response amplitudes to blade mistuning is reported. This work reports one of the first systematic experiments carried out to demonstrate and quantify the effect of mistuning on the forced response of bladed disks.

Forced Response of Bladed Disks in Cyclic Symmetry With Underplatform Dampers

2006 ◽  
Author(s):  
Stefano Zucca ◽  
Juan Borrajo ◽  
Muzio M. Gola
Keyword(s):  
Contact Stiffness ◽  
Harmonic Balance Method ◽  
Rigid Bodies ◽  
Forced Response ◽  
Numerical Code ◽  
Algebraic Equations ◽  
Bladed Disks ◽  
Disk Model ◽  
Normal Contact ◽  
Linear Algebraic Equations

In this paper a methodology for forced response calculation of bladed disks with underplatform dampers is described. The FE disk model, supposed to be cyclically symmetric, is reduced by means of Component Mode Synthesis and then DOFs lying at interfaces are further reduced by means of interface modes. Underplatform dampers are modeled as rigid bodies translating both in the radial and in the tangential direction of the engine. Contacts between blade platforms and damper are simulated by means of contact elements characterized by both tangential and normal contact stiffness, allowing partial separation of contact surfaces. Differential equilibrium equations are turned in non-linear algebraic equations by means of the Harmonic Balance Method (HBM). The methodology is implemented in a numerical code for forced response calculation of frictionally damped bladed disks. Numerical calculations are performed to evaluate the effectiveness of both the reduced order model and the underplatform model in simulating the dynamic behavior of bladed disks in presence of underplatform dampers.

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