Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry

Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry

with Applications to Turbomachinery

MOTIVATION & BACKGROUND

MotivationBladed Disk Assemblies

MotivationEngine Order Excitation

Resonance Structure / Conditions for ResonanceMotivation

Order-Tuned AbsorbersMotivation

MotivationVibration Reduction via Order-Tuned Absorbers

Motivation

Tuned Dampers

Sleeves

Chamber & End Caps

Vibration Reduction via Order-Tuned Absorbers

MotivationVibration Reduction via Order-Tuned Absorbers

Vibration Reduction via Order-Tuned AbsorbersMotivation

1. Quantify/understand underlying linear resonance structure;

2. Design absorbers to eliminate/reduce blade vibrations; and

3. Generalize to include effects of nonlinearity.

Goals of this WorkMotivation

– How does Campbell diagram representation change when order-tuned absorbers are present?

– Can nonlinearity be exploited to further improve the linear design?

– Exploit underlying linear resonance structure for linear absorber design.

Outline1. Motivation and Background

Frequency- and Order-Tuned AbsorbersCyclic SystemsTheory of Circulants / Mathematical PreliminariesEngine Order Excitation

2. The Linear AnalysisModel / FormulationModal Analysis / Forced ResponseLinear Resonance Structure / Absorber TuningEffects of Damping

3. The Nonlinear AnalysisMathematical Model / Path SelectionFormulation: Scaling / AveragingTraveling Wave Forced Response / StabilityNonlinear Absorber Tuning

4. ConclusionsRecommendations for Absorber DesignSummary of ContributionsDirections for Future Work

BackgroundEngine Order Excitation

BackgroundEngine Order Excitation

BackgroundEngine Order Excitation

BackgroundEngine Order Excitation

BackgroundEngine Order Excitation

Model / FormulationModal Analysis / Forced Response

Linear Resonance Structure / Absorber TuningEffects of Damping

Summary

Mathematical Model

THE LINEAR ANALYSISand

Mathematical ModelBladed Disk Assembly with Absorbers

Mathematical ModelLinearized System Model

Modal AnalysisBlock Decoupling the EOM

Modal AnalysisSteady-State Modal Response

Modal AnalysisSteady-State Modal Response

Special Cases1

1. Blades Locked, Absorbers Free

– Gives Linear Absorber Tuning Order

2. Blades Free, Absorbers Locked

– A Benchmark to evaluate absorber performance

3. Single Isolated Sector, Blade/Absorber Free

– Demonstrates the essential features of the full coupled system

Special Cases1Blades Locked, Absorbers Free

( )

Special Cases2Blades Free, Absorbers Locked

( )

Special Cases2Blades Free, Absorbers Locked

( )

Special Cases3Single Isolated Sector, Blade/Absorber Free

( )

Special Cases3Single Isolated Sector, Blade/Absorber Free

( )

Special Cases3Single Isolated Sector, Blade/Absorber Free

( )

Special Cases3Single Isolated Sector, Blade/Absorber Free

( )

Special Cases3Single Isolated Sector, Blade/Absorber Free

( )

Special Cases3Single Isolated Sector, Blade/Absorber Free

( )

N = 10 Sectors, Blades/Absorbers FreeLinear Resonance Structure

The Effects of Detuning: No-Resonance Gap Linear Resonance Structure

Linear Forced Response

Absorbers FreeLinear Forced Response

Absorbers FreeLinear Forced Response

The Effects of Damping

Linear Absorber DesignSummary

Linear Absorber Design

• Absorbers are effective– No-resonance zone*– Ideal tuning

(Complete reduction of blade motions**)

– Slight undertuning (Good reduction of blade motions and no resonances over full range of rotor speeds)

Summary

* Persists in the presence of sufficiently small damping** No absorber damping, independent of blade damping

Linear Absorber Design

* Persists in the presence of sufficiently small damping** No absorber damping, independent of blade damping

Summary

Recommendation

• Absorbers are effective– No-resonance zone*– Ideal tuning

(Complete reduction of blade motions**)

– Slight undertuning (Good reduction of blade motions and no resonances over full range of rotor speeds)

Mathematical Model / Path SelectionFormulation: Scaling / AveragingTW Forced Response / Stability

NL Absorber TuningSummary

THE NONLINEAR ANALYSIS

Mathematical ModelNonlinear Sector

Mathematical ModelAbsorber Path

FormulationScaled Sector Models

Linear Resonance Structure of the Scaled SystemFormulation

Linear Resonance Structure of the Scaled SystemFormulation

FormulationAveraged Sector Models

FormulationAveraged Sector Models

FormulationAveraged Sector Models

Features of the Forced Response

Consider Separately:

• Isolated Nonlinear System

• Coupled Nonlinear System

– Embodies the basic NL features, except certain stability results

– Predicts instabilities of the desired TW response

The Isolated Nonlinear SystemFrequency Response

Critical Nonlinear Tuning

– Highly sensitive to parameter uncertainty – Depends on rotor speed and force amplitude– For proper linear undertuning ( 0) requires undesirable

hardening absorber path

The Isolated Nonlinear System

Traveling Wave ResponseThe Coupled Nonlinear System

Possible Symmetry-Breaking BifurcationsThe Coupled Nonlinear System

Possible Symmetry-Breaking BifurcationsThe Coupled Nonlinear System

Frequency ResponseThe Coupled Nonlinear System

Nonlinear Absorber Design

– No-resonance zone persists– Nonlinearity cannot be exploited to improve performance– Softening paths desirable / hardening paths undesirable– No instabilities of the desired TW response

Summary

Recommendations for Absorber DesignSummary of ContributionsDirections for Future Work

Acknowledgments

CONCLUSIONS

ConclusionsRecommendations for Absorber Design

• Summary of Contributions– First systematic analytical study of its kind – Existence of a no-resonance zone– First-order nonlinear effects– No instabilities to non-traveling-wave responses found

• Absorber Design Recommendations– Select linear detuning within the no-resonance gap– Keep absorber motions as linear as possible– If nonlinearity is unavoidable, softening characteristics are desirable

• Directions for Future Work– Higher-fidelity blade models– Mistuning studies– Experimental validation

Conclusions

• The National Science Foundation– Grant CMS-0408866

Acknowledgments

• The National Science Foundation– Grant CMS-0408866

• Doctoral Committee– Steve Shaw (advisor)– Christophe Pierre [ Matt Castanier ]– Alan Haddow– Brian Feeny– Cevat Gokcek– Hassan Khalil

Acknowledgments

• The National Science Foundation– Grant CMS-0408866

• Doctoral Committee– Steve Shaw (advisor)– Christophe Pierre [ Matt Castanier ]– Alan Haddow– Brian Feeny– Cevat Gokcek– Hassan Khalil

• Colleagues– Jeff Rhoads– Pat Staron– …

Acknowledgments

• The National Science Foundation– Grant CMS-0408866

• Doctoral Committee– Steve Shaw (advisor)– Christophe Pierre [ Matt Castanier ]– Alan Haddow– Brian Feeny– Cevat Gokcek– Hassan Khalil

• Colleagues– Jeff Rhoads– Pat Staron– …

• Family and Friends– Julie Olson– Bob & Viv Olson, Gregg McFarlyn– Jim Coughlin

Acknowledgments

• The National Science Foundation– Grant CMS-0408866

• Doctoral Committee– Steve Shaw (advisor)– Christophe Pierre [ Matt Castanier ]– Alan Haddow– Brian Feeny– Cevat Gokcek– Hassan Khalil

• Colleagues– Jeff Rhoads– Pat Staron– …

• Family and Friends– Julie Olson– Bob & Viv Olson, Gregg McFarlyn– Jim Coughlin

• Donald E. Knuth, Leslie Lamport– Creators of the TeX and LaTeX systems for typesetting

Acknowledgments