Rotating Cyclic Systems with Order-Tuned Vibration Absorbers

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Rotating Cyclic Systems with Order-Tuned
Vibration Absorbers
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Outline

• Cyclic Structures
• Order-Tuned Absorbers
• Motivation Background
• The Linear Problem
• The Nonlinear Problem
• Conclusions Future Work

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Relevant Previous Work

• Order-Tuned Vibration Absorbers
• Den Hartog, Denman, Cronin, Shaw, Borowski,
  Duffy,
• Vibration Characteristics of Bladed Disk
  Assemblies
• Ewins, Srinivasan, Griffin, Whitehead, Pierre,
• Localization
• Pierre, Bajaj, Vakakis,
• Linear Cyclic Systems
• Pierre, Shapiro, Bajaj, Vakakis,
• Nonlinear Cyclic Systems
• Bajaj, Vakakis, Coller, King,

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Background
Bladed Disk Assemblies
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Background
Engine Order Excitation
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Background
Order Excitation
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Background
Self-tuning Impact Damper
Turbine Blade
Sleeves
Tuned Dampers
Chamber End Caps
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Order-Tuned Vibration Absorbers
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Order-Tuned Vibration Absorbers
Torsional Vibration Reduction
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History Torsional Vibration Reduction

• Early designs
• Sizing, physical arrangement
• Linear tuning local path curvature - 1930
• Implementations
• Light aircraft engines, WWII
• Helicopter rotors,1980s
• Experimental/racing automotive engines,1990-
• Path designs for nonlinearities
• Cycloids (Madden, 1980), Epicycloids (Denman,
  1991), Subharmonic epicycloids (Lee Shaw,
  1995), General paths (Alsuwayian and Shaw, 2001)

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Absorber Paths
General Path Representation
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Absorber Paths

• Linear Tuning
• Frequency of small amplitude motions
• Circles
• Easily manufactured
• Strong nonlinear effects, softening,
• Cycloids
• The tautochrone in uniform fields
• Weak nonlinear effects, hardening,
• Epicycloid
• The tautochrone in radial fields
• Linear absorber motions at all amplitudes,

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Mathematical Model
Equations of Motion
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Mathematical Model
Equations of Motion
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Mathematical Model
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Cyclic Symmetry
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The Linearized System
Sector Model
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The Linearized System
System Model M DOF/Sector
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The Linearized System
System Model M DOF/Sector
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Mathematical Preliminaries
Circulant Matrices
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Mathematical Preliminaries
Diagonalization of a Block Circulant
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Mathematical Preliminaries
The Fourier Matrix
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Mathematical Preliminaries
The Direct (Kronecker) Product
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Linear Vibration
(Block) Decoupling the EOM
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Linear Free Vibration
One DOF/Sector
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Linear Free Vibration
One DOF/Sector
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Linear Free Vibration
One DOF/Sector
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Linear Forced Vibration
Steady-State Response
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Linear Forced Vibration
Steady-State Physical Response
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Linear Forced Vibration
Blade Response (Absorbers Locked)
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Linear Isolated Absorber Response
Absorber Free, Blades Locked
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Linear Response
N Blades with Absorbers
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Linear Response
The Effects of Detuning, Weak Coupling (like N1)
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Linear Response
The Effects of Detuning, Strong Coupling
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Linear Response
The Effects of Detuning
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Linear Response
Frequency Response (zero damping)
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Nonlinear Blade Response
One DOF/Sector (Blades) Weakly Nonlinear
Strong Coupling
Weak Coupling
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Nonlinear Blade Response
One DOF/Sector (Blades) Strongly Coupled
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Nonlinear Blade Response
One DOF/Sector (Blades) Strongly Coupled
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Nonlinear Blade Response
One DOF/Sector (Blades) Strongly Coupled
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Nonlinear Blade Response
One DOF/Sector (Blades) Strongly Coupled
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Nonlinear Blade Response
One DOF/Sector (Blades) Weakly Coupled
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Nonlinear Blade Response
One DOF/Sector (Blades) Weakly Coupled
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Nonlinear Blade Response
One DOF/Sector (Blades) Weakly Coupled
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Nonlinear Blade Response
One DOF/Sector (Blades) Weakly Coupled
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Nonlinear Blade Response
One DOF/Sector (Blades) Weakly Coupled
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Nonlinear Blade Response
One DOF/Sector (Blades) Weakly Nonlinear
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Linear Blade Nonlinear Absorber
Assumptions and Scaling
Goal Capture nonlinear absorber behavior
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Linear Blade Nonlinear Absorber
N Blade/Absorbers, Weak Coupling
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Linear Blade Nonlinear Absorber
N Blade/Absorbers, Weak Coupling
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Linear Blade Nonlinear Absorber
Weak Coupling
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Linear Blade Nonlinear Absorber
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Linear Blade Nonlinear Absorber
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Linear Blade Nonlinear Absorber
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Summary Conclusions

• Linear System, Blades Absorbers absorber
  effective, no resonance zone
• Nonlinear System, Blades Only traveling wave
  excitation limits some types of instabilities
• Nonlinear System, Blades Absorbers absorbers
  can be effective, but nonlinear absorber paths
  often lead to system resonance

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Directions for Future Work

• Linear System effects of damping, mistuning
• Nonlinear System, blades only post-bifurcation
  analysis
• Nonlinear System, blades absorbers scaling
  for tautochronic (linear) absorber path,
  detailed parameter studies
• Mistuning random and intentional reduction or
  elimination of symmetry
• Experiments