Order-Tuned Vibration Absorbers for Cyclic Rotating Flexible Structures
Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry
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Transcript of 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
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Special Cases2Blades Free, Absorbers Locked
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Special Cases2Blades Free, Absorbers Locked
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Special Cases3Single Isolated Sector, Blade/Absorber Free
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Special Cases3Single Isolated Sector, Blade/Absorber Free
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Special Cases3Single Isolated Sector, Blade/Absorber Free
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Special Cases3Single Isolated Sector, Blade/Absorber Free
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Special Cases3Single Isolated Sector, Blade/Absorber Free
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Special Cases3Single Isolated Sector, Blade/Absorber Free
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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