Academic Blade Geometries for Baseline Comparisons of Industry-Specific Forced Response Predictions

2017 ◽  
Author(s):  
James H. Little ◽  
Jeffrey L. Kauffman ◽  
Matthias Huels
Keyword(s):  
Model Reduction ◽  
Turbine Blades ◽  
Reduction Process ◽  
Forced Response ◽  
Vibration Characteristics ◽  
Contact Dynamics ◽  
Qualitative Behavior ◽  
Friction Damper ◽  
Blade Vibration ◽  
Reduced Order

Predicting the energy dissipation associated with contact of underplatform dampers remains a critical challenge in turbomachinery blade and friction damper design. Typical turbomachinery blade forced vibration response analyses rely on reduced order models and simplified nonlinear codes to predict blade vibration characteristics in a computationally tractable manner. Recent research has focused on both the model reduction process and simulation of the contact dynamics. This paper proposes two academic turbine blade geometries with coupled underplatform dampers as vehicles by which these model reduction and forced response simulation techniques may be compared. The blades correspond to two types of freestanding turbine blades and demonstrate the same qualitative behavior as more complex industry geometries. The blade geometries are fully described here and analyzed using the same procedure as used for an industry-specific blade. Standard results are presented in terms of resonance frequency, amplitude, and damping across a range of aerodynamic excitation. In addition, the predicted blade vibration characteristics are examined under variations in the contact interface: friction coefficient, damper / platform surface roughness, and damper mass, with relative sensitivities to each term generated. Finally, the effect of the number of modes retained in the reduced order model is studied to uncover patterns of convergence as well as to provide additional sets of standard data for comparison with other model reduction and forced response simulation methods.

A Resonance Tracking Algorithm for the Prediction of Turbine Forced Response With Friction Dampers

2000 ◽  
Author(s):  
C. Bréard ◽  
J. S. Green ◽  
M. Vahdati ◽  
M. Imregun
Keyword(s):  
Rotor Blade ◽  
Turbine Blades ◽  
Turbine Rotor ◽  
Forced Response ◽  
Rotor Blades ◽  
Test Case ◽  
Friction Damper ◽  
Blade Vibration ◽  
Frequency Shifts ◽  
Friction Dampers

This paper presents an iterative method for determining the resonant speed shift when non-linear friction dampers are included in turbine blade roots. Such a need arises when conducting response calculations for turbine blades where the unsteady aerodynamic excitation must be computed at the exact resonant speed of interest. The inclusion of friction dampers is known to raise the resonant frequencies by up to 20% from the standard assembly frequencies. The iterative procedure uses a viscous, time-accurate flow representation for determining the aerodynamic forcing, a look-up table for evaluating the aerodynamic boundary conditions at any speed, and a time-domain friction damping module for resonance tracking. The methodology was applied to an HP turbine rotor test case where the resonances of interest were due to the 1T and 2F blade modes under 40 engine-order excitation. The forced response computations were conducted using a multi-stage approach in order to avoid errors associated with “linking” single stage computations since the spacing between the two bladerows was relatively small. Three friction damper elements were used for each rotor blade. To improve the computational efficiency, the number of rotor blades was decreased by 2 to 90 in order to obtain a stator/rotor blade ratio of 4/9. However, the blade geometry was skewed in order to match the capacity (mass flow rate) of the components and the condition being analysed. Frequency shifts of 3.2% and 20.0% were predicted for the 1T/40EO and 2F/40EO resonances in about 3 iterations. The predicted frequency shifts and the dynamic behaviour of the friction dampers were found to be within the expected range. Furthermore, the measured and predicted blade vibration amplitudes showed a good agreement, indicating that the methodology can be applied to industrial problems.

A Physics-Based Dynamic Model for Boilers: Part 2 — Model Reduction in a Cogeneration Application

2017 ◽  
Author(s):  
Matthew J. Blom ◽  
Michael J. Brear ◽  
Chris G. Manzie ◽  
Ashley P. Wiese
Keyword(s):  
Model Reduction ◽  
Gas Turbine ◽  
Dynamic Model ◽  
Reduction Process ◽  
Singular Perturbation Theory ◽  
Reduced Order Models ◽  
Reduced Order ◽  
One Dimensional ◽  
Time Scale Separation ◽  
Modelling Framework

This paper is the second part of a two part study that develops, validates and integrates a one-dimensional, physics-based, dynamic boiler model. Part 1 of this study [1] extended and validated a particular modelling framework to boilers. This paper uses this framework to first present a higher order model of a gas turbine based cogeneration plant. The significant dynamics of the cogeneration system are then identified, corresponding to states in the gas path, the steam path, the gas turbine shaft, gas turbine wall temperatures and boiler wall temperatures. A model reduction process based on time scale separation and singular perturbation theory is then demonstrated. Three candidate reduced order models are identified using this model reduction process, and the simplest, acceptable dynamic model of this integrated plant is found to require retention of both the gas turbine and boiler wall temperature dynamics. Subsequent analysis of computation times for the original physics-based one-dimensional model and the candidate, reduced order models demonstrates that significantly faster than real time simulation is possible in all cases. Furthermore, with systematic replacement of the algebraic states with feedforward maps in the reduced order models, further computational savings of up to one order of magnitude can be achieved. This combination of model fidelity and computational tractability suggest suggests that the resulting reduced order models may be suitable for use in model based control of cogeneration plants.

Alternate Mistuning of Turbine Bladings Coupled by Underplatform Dampers

2012 ◽  
Author(s):  
S. Tatzko ◽  
L. Panning-von Scheidt ◽  
J. Wallaschek ◽  
A. Kayser
Keyword(s):  
Turbine Blades ◽  
Steady State Solution ◽  
Computational Effort ◽  
Forced Response ◽  
Nonlinear Coupling ◽  
Cyclic Symmetry ◽  
Periodic Excitation ◽  
Blade Vibration ◽  
Mode Localization ◽  
Bladed Disk

In turbo machinery design it is important to avoid vibrations that can destroy the turbine in the last resort. The rotating structure is exposed to periodic excitation forces. Two main types of periodic excitation can be distinguished. Flutter is the effect when mass flow forces couple with a natural vibration mode. The result is a negative damping coefficient and amplitudes will rise up to malfunction of the structure. The engine order excitation is a periodic excitation where the force signal is directly related to the speed of the rotor. A forced response calculation gives information about the blade vibration. Nonlinear coupling, i.e. friction coupling, between blades is used to increase damping of the bladed disk. Dynamic analysis of turbine blades with nonlinear coupling is a complex task and computer simulations are inevitable. Various techniques have been developed to reduce computational effort. The cyclic symmetry approach assumes each blade around the disk to be identical. Thus only one sector of the disk is sufficient to compute the steady state solution of the whole turbine blading. However, it has been observed that mistuning of blades reduces the flutter instability. On the other hand statistical mistuning can lead to dangerously high forced response amplitudes due to mode localization. A compromise is intentional mistuning. The simplest approach is alternate mistuning with every other blade exhibiting identical mechanical properties. This work explains in detail how a turbine bladed disk can be modeled when alternate mistuning is applied intentionally. Cyclic symmetry is used and each sector comprises two blades. This untypical choice of the sector size has significant impact on results of a cyclic modal analysis. Simulation results show the influence of alternate mistuned turbine bladings which are coupled by underplatform damper elements.

Fatigue Damage Prevention on Turbine Blades: Study of Underplatform Damper Shape

2007 ◽  
Vol 347 ◽  
pp. 159-164 ◽  
Author(s):  
Teresa Berruti ◽  
Christian M. Firrone ◽  
M. Pizzolante ◽  
Muzio M. Gola
Keyword(s):  
Turbine Blades ◽  
Forced Vibrations ◽  
Forced Response ◽  
Numerical Code ◽  
Vibration Energy ◽  
Test Rig ◽  
Blade Vibration ◽  
Static Test ◽  
Underplatform Damper ◽  
Contact Parameters

Forced vibrations can lead to an irreparable damage of a blade array. Devices called “underplatform damper” that dissipate the vibration energy are employed in order to reduce blade vibration amplitude. The present paper deals with the design of the underplatform damper. A numerical code to calculate the forced response of a blade array with dampers has been previously purposely developed. A method is here proposed for the estimation of the unknown contact parameters demanded by the code. The computation results are here validated by means of comparison with experimental results on a static test rig. Three dampers with different shape are tested.

A Method to Reduce the Energy Localization in Mistuned Bladed Disks by Application-Specific Blade Pattern Arrangement

2016 ◽  
Vol 138 (9) ◽  
Author(s):  
Andreas Hohl ◽  
Jörg Wallaschek
Keyword(s):  
Turbine Blades ◽  
Forced Response ◽  
Energy Localization ◽  
Order Model ◽  
Small Deviations ◽  
Reduced Order ◽  
Bladed Disk ◽  
Increased Risk ◽  
Manufacturing Tolerances ◽  
Application Specific

The focus of the paper is the analysis of mistuning, which are small deviations of the blade properties, e.g., due to manufacturing tolerances. The resonant amplitudes of turbine blades are very sensitive to these deviations which can lead to significantly increased vibratory response of some blades with the increased risk of high cycle fatigue. The main part of the paper discusses about the methods that can be used to find blade patterns which are insensitive to energy localization. The sensitivity of the forced response against harmonic mistuning, which is a harmonic alignment of the blades respective to the mistuning factors of the single blades, is examined. A previously developed reduced order model (ROM) is used to efficiently conduct Monte Carlo simulations (MCSs). Especially the influence of the variance of the harmonically mistuned blade patterns is discussed. On the basis of this analysis, rules are developed to suppress the energy localization. The rules are mainly focused on the alignment of the blades around the bladed disk. The approach also takes advantage of the special properties of harmonic mistuning patterns. An assignment of the blades to insensitive harmonic mistuning patterns with a specific variance and number of periods is used to reduce the maximum of the amplification factor of the forced response. A similar approach uses intentional mistuning patterns with different blade types which are aligned harmonically and are insensitive to an additional stochastic mistuning. In case of specific combinations of the dependent parameters, especially the variance of the mistuning factors and the number of periods, the energy localization can be reduced considerably.

scholarly journals Input–output measures for model reduction and closed-loop control: application to global modes

2011 ◽  
Vol 685 ◽  
pp. 23-53 ◽  
Author(s):  
Alexandre Barbagallo ◽  
Denis Sipp ◽  
Peter J. Schmid
Keyword(s):  
Feedback Control ◽  
Model Reduction ◽  
Degrees Of Freedom ◽  
Closed Loop ◽  
Reduction Process ◽  
Open Loop ◽  
Input Output ◽  
Control Applications ◽  
Reduced Order ◽  
Global Modes

AbstractFeedback control applications for flows with a large number of degrees of freedom require the reduction of the full flow model to a system with significantly fewer degrees of freedom. This model-reduction process is accomplished by Galerkin projections using a reduction basis composed of modal structures that ideally preserve the input–output behaviour between actuators and sensors and ultimately result in a stabilized compensated system. In this study, global modes are critically assessed as to their suitability as a reduction basis, and the globally unstable, two-dimensional flow over an open cavity is used as a test case. Four criteria are introduced to select from the global spectrum the modes that are included in the reduction basis. Based on these criteria, four reduced-order models are tested by computing open-loop (transfer function) and closed-loop (stability) characteristics. Even though weak global instabilities can be suppressed, the concept of reduced-order compensators based on global modes does not demonstrate sufficient robustness to be recommended as a suitable choice for model reduction in feedback control applications. The investigation also reveals a compelling link between frequency-restricted input–output measures of open-loop behaviour and closed-loop performance, which suggests the departure from mathematically motivated ${\mathscr{H}}_{\infty } $-measures for model reduction toward more physically based norms; a particular frequency-restricted input–output measure is proposed in this study which more accurately predicts the closed-loop behaviour of the reduced-order model and yields a stable compensated system with a markedly reduced number of degrees of freedom.

scholarly journals Sampling-free parametric model reduction of systems with structured parameter variation

2018 ◽  
Author(s):  
Christopher Beattie ◽  
Serkan Gugercin ◽  
Zoran Tomljanović
Keyword(s):  
Model Reduction ◽  
Transfer Functions ◽  
Linear Time ◽  
Mass Matrix ◽  
Reduction Process ◽  
Parametric Model ◽  
Reduced Order Model ◽  
Computationally Efficient ◽  
Order Model ◽  
Reduced Order

We consider a parametric linear time invariant dynamical systems represented in state-space form as $$E \dot x(t) = A(p) x(t) + Bu(t), \\ y(t) = Cx(t),$$ where $E, A(p) \in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m} $ and $C\in \mathbb{R}^{l\times n}$. Here $x(t)\in \mathbb{R}^{n} $ denotes the state variable, while $u(t)\in \mathbb{R}^{m}$ and $y(t)\in \mathbb{R}^{l}$ represent, respectively, the inputs and outputs of the system. We assume that $A(p)$ depends on $k\ll n$ parameters $p=(p_1, p_2, \ldots, p_k)$ such that we may write $$A(p)=A_0+U\,\diag (p_1, p_2, \ldots, p_k)V^T,$$ where $U, V \in \mathbb{R}^{n\times k}$ are given fixed matrices.We propose an approach for approximating the full-order transfer function $H(s;p)=C(s E -A(p))^{-1}B$ with a reduced-order model that retains the structure of parametric dependence and (typically) offers uniformly high fidelity across the full parameter range. Remarkably, the proposed reduction process removes the need for parameter sampling and thus does not depend on identifying particular parameter values of interest. Our approach is based on the classic Sherman-Morrison-Woodbury formula and allows us to construct a parameterized reduced order model from transfer functions of four subsystems that do not depend on parameters, allowing one to apply well-established model reduction techniques for non-parametric systems. The overall process is well suited for computationally efficient parameter optimization and the study of important system properties. One of the main applications of our approach is for damping optimization: we consider a vibrational system described by $$ \begin{equation}\label{MDK} \begin{array}{rl} M\ddot q(t)+(C_{int} + C_{ext})\dot q(t)+Kq(t)&=E w(t),\\ z(t)&=Hq(t), \end{array} \end{equation} $$ where the mass matrix, $M$, and stiffness matrix, $K$, are real, symmetric positive-definite matrices of order $n$. Here, $q(t)$ is a vector of displacements and rotations, while $ w(t) $ and $z(t) $ represent, respectively, the inputs (typically viewed as potentially disruptive) and outputs of the system. Damping in the structure is modeled as viscous damping determined by $C_{int} + C_{ext}$ where $C_{int}$ and $C_{ext}$ represent contributions from internal and external damping, respectively. Information regarding damper geometry and positioning as well as the corresponding damping viscosities are encoded in $C_{ext}= U\diag{(p_1, p_2, \ldots, p_k)} U^T$ where $U \in \mathbb{R}^{n\times k}$ determines the placement and geometry of the external dampers. The main problem is to determine the best damping matrix that is able to minimize influence of the disturbances, $w$, on the output of the system $z$. We use a minimization criteria based on the $\mathcal{H}_2$ system norm. In realistic settings, damping optimization is a very demanding problem. We find that the parametric model reduction approach described here offers a new tool with significant advantages for the efficient optimization of damping in such problems.

A Method to Reduce the Energy Localization in Mistuned Bladed Disks by Application-Specific Blade Pattern Arrangement

2015 ◽  
Author(s):  
Andreas Hohl ◽  
Jörg Wallaschek
Keyword(s):  
Turbine Blades ◽  
Forced Response ◽  
Energy Localization ◽  
Order Model ◽  
Small Deviations ◽  
Reduced Order ◽  
Bladed Disk ◽  
Increased Risk ◽  
Manufacturing Tolerances ◽  
Application Specific

The focus of the paper is the analysis of mistuning, that is small deviations of the blade properties e.g. due to manufacturing tolerances. The resonant amplitudes of turbine blades are very sensitive to these deviations which can lead to significantly increased vibratory response of some blades with increased risk of high cycle fatigue. The main part of the paper discusses methods that can be used to find blade patterns which are insensitive to energy localization. The sensitivity of the forced response against harmonic mistuning, that is a harmonic alignment of the blades respective to the mistuning factors of the single blades, is examined. A previously developed reduced order model is used to efficiently conduct Monte Carlo Simulations (MCS). Especially the influence of the variance of the harmonically mistuned blade patterns is discussed. On the basis of this analysis rules are developed to suppress the energy localization. The rules are mainly focused on the alignment of the blades around the bladed disk. The approach also takes advantage of the special properties of harmonic mistuning patterns. An assignment of the blades to insensitive harmonic mistuning patterns with a specific variance and number of periods is used to reduce the maximum of the amplification factor of the forced response. A similar approach uses intentional mistuning patterns with different blade types which are aligned harmonically and are insensitive to an additional stochastic mistuning. In case of specific combinations of the dependent parameters, especially the variance of the mistuning factors and the number of periods, the energy localization can be reduced considerably.

scholarly journals Investigations on the Blade Vibration of a Radial Inflow Micro Gas Turbine Wheel

2007 ◽  
Vol 2007 ◽  
pp. 1-10 ◽  
Author(s):  
Shijie Guo
Keyword(s):  
Gas Turbine ◽  
Young’S Modulus ◽  
Young's Modulus ◽  
Turbine Blades ◽  
Pressure Fluctuations ◽  
Leading Edge ◽  
Coupled Modes ◽  
Blade Vibration ◽  
Turbine Wheel ◽  
Micro Gas Turbine

This paper demonstrates the investigations on the blade vibration of a radial inflow micro gas turbine wheel. Firstly, the dependence of Young's modulus on temperature was measured since it is a major concern in structure analysis. It is demonstrated that Young's modulus depends on temperature greatly and the dependence should be considered in vibration analysis, but the temperature gradient from the leading edge to the trailing edge of a blade can be ignored by applying the mean temperature. Secondly, turbine blades suffer many excitations during operation, such as pressure fluctuations (unsteady aerodynamic forces), torque fluctuations, and so forth. Meanwhile, they have many kinds of vibration modes, typical ones being blade-hub (disk) coupled modes and blade-shaft (torsional, longitudinal) coupled modes. Model experiments and FEM analysis were conducted to study the coupled vibrations and to identify the modes which are more likely to be excited. The results show that torque fluctuations and uniform pressure fluctuations are more likely to excite resonance of blade-shaft (torsional, longitudinal) coupled modes. Impact excitations and propagating pressure fluctuations are more likely to excite blade-hub (disk) coupled modes.

scholarly journals Improved Reliability of Bladed Disks due to Friction Dampers

1997 ◽  
Author(s):  
Walter Sextro ◽  
Karl Popp ◽  
Ivo Wolter
Keyword(s):  
Dry Friction ◽  
Three Dimensional ◽  
Line Contact ◽  
Two Dimensional ◽  
Centrifugal Forces ◽  
Friction Damper ◽  
Blade Vibration ◽  
Friction Dampers ◽  
Bladed Disk ◽  
Bladed Disk Assembly

Friction dampers are installed underneath the blade platforms to improve the reliability. Because of centrifugal forces the dampers are pressed onto the platforms. Due to dry friction and the relative motion between blades and dampers, energy is dissipated, which results in a reduction of blade vibration amplitudes. The geometry of the contact is in many cases like a Hertzian line contact. A three-dimensional motion of the blades results in a two-dimensional motion of one contact line of the friction dampers in the contact plane. An experiment with one friction damper between two blades is used to verify the two-dimensional contact model including microslip. By optimizing the friction dampers masses, the best damping effects are obtained. Finally, different methods are shown to calculate the envelope of a three-dimensional response of a detuned bladed disk assembly (V84.3-4th-stage turbine blade) with friction dampers.

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