Order-Tuned Vibration Absorbers for Systems with Cyclic Symmetry

1
Order-Tuned Vibration Absorbers for Systems with
Cyclic Symmetry
with Applications to Turbomachinery
2
MOTIVATION BACKGROUND
3
Motivation
Bladed Disk Assemblies
4
Motivation
Engine Order Excitation
5
Motivation
Resonance Structure / Conditions for Resonance
6
Motivation
Order-Tuned Absorbers
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Motivation
Vibration Reduction via Order-Tuned Absorbers
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Motivation
Vibration Reduction via Order-Tuned Absorbers
Sleeves
Tuned Dampers
Chamber End Caps
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Motivation
Vibration Reduction via Order-Tuned Absorbers
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Motivation
Vibration Reduction via Order-Tuned Absorbers
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Motivation
Goals of this Work

1. Quantify/understand underlying linear resonance
   structure
2. Design absorbers to eliminate/reduce blade
   vibrations and
3. Generalize to include effects of nonlinearity.

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

• Exploit underlying linear resonance structure for
  linear absorber design.

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

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Outline

• Motivation and Background
• Frequency- and Order-Tuned Absorbers
• Cyclic Systems
• Theory of Circulants / Mathematical Preliminaries
• Engine Order Excitation
• The Linear Analysis
• Model / Formulation
• Modal Analysis / Forced Response
• Linear Resonance Structure / Absorber Tuning
• Effects of Damping
• The Nonlinear Analysis
• Mathematical Model / Path Selection
• Formulation Scaling / Averaging
• Traveling Wave Forced Response / Stability
• Nonlinear Absorber Tuning
• Conclusions
• Recommendations for Absorber Design
• Summary of Contributions
• Directions for Future Work

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Background
Engine Order Excitation
14
Background
Engine Order Excitation
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Background
Engine Order Excitation
16
Background
Engine Order Excitation
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Background
Engine Order Excitation
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Model / Formulation Modal Analysis / Forced
Response Linear Resonance Structure / Absorber
Tuning Effects of Damping Summary
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Mathematical Model
Bladed Disk Assembly with Absorbers
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Mathematical Model
Linearized System Model
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Modal Analysis
Block Decoupling the EOM
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Modal Analysis
Steady-State Modal Response
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Modal Analysis
Steady-State Modal Response
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Special Cases1

• Blades Locked, Absorbers Free
• Gives Linear Absorber Tuning Order
• Blades Free, Absorbers Locked
• A Benchmark to evaluate absorber performance
• Single Isolated Sector, Blade/Absorber Free
• Demonstrates the essential features of the full
  coupled system

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Special Cases1
( )
Blades Locked, Absorbers Free
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Special Cases2
( )
Blades Free, Absorbers Locked
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Special Cases2
( )
Blades Free, Absorbers Locked
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Special Cases3
( )
Single Isolated Sector, Blade/Absorber Free
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Special Cases3
( )
Single Isolated Sector, Blade/Absorber Free
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Special Cases3
( )
Single Isolated Sector, Blade/Absorber Free
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Special Cases3
( )
Single Isolated Sector, Blade/Absorber Free
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Special Cases3
( )
Single Isolated Sector, Blade/Absorber Free
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Special Cases3
( )
Single Isolated Sector, Blade/Absorber Free
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Linear Resonance Structure
N 10 Sectors, Blades/Absorbers Free
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Linear Resonance Structure
The Effects of Detuning No-Resonance Gap
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Linear Forced Response
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Linear Forced Response
Absorbers Free
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Linear Forced Response
Absorbers Free
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The Effects of Damping
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Linear Absorber Design
Summary
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Linear Absorber Design
Summary

• 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)

Persists in the presence of sufficiently small
damping No absorber damping, independent of
blade damping
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Linear Absorber Design
Summary

• 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)

Recommendation
Persists in the presence of sufficiently small
damping No absorber damping, independent of
blade damping
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THE NONLINEAR ANALYSIS
Mathematical Model / Path Selection Formulation
Scaling / Averaging TW Forced Response /
Stability NL Absorber Tuning Summary
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Mathematical Model
Nonlinear Sector
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Mathematical Model
Absorber Path
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Formulation
Scaled Sector Models
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Formulation
Linear Resonance Structure of the Scaled System
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Formulation
Linear Resonance Structure of the Scaled System
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Formulation
Averaged Sector Models
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Formulation
Averaged Sector Models
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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

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The Isolated Nonlinear System
Frequency Response
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The Isolated Nonlinear System
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

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The Coupled Nonlinear System
Traveling Wave Response
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The Coupled Nonlinear System
Possible Symmetry-Breaking Bifurcations
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The Coupled Nonlinear System
Possible Symmetry-Breaking Bifurcations
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The Coupled Nonlinear System
Frequency Response
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Nonlinear Absorber Design
Summary

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

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CONCLUSIONS
Recommendations for Absorber Design Summary of
Contributions Directions for Future
Work Acknowledgments
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Conclusions
Recommendations for Absorber Design
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Conclusions

• 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

63
Acknowledgments

• The National Science Foundation
• Grant CMS-0408866

64
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

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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

66
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

67
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

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