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Vibration Absorbers for Cyclic Rotating Flexible Structures

Vibration Absorbers for Cyclic Rotating Flexible Structures. Linear and Nonlinear Tuning. Brian J. Olson Senior Staff Engineer The Johns Hopkins University Applied Physics Laboratory. Steven W. Shaw Professor Michigan State University. SMASIS08_5-7, Ellicott City, MD, October 28-30, 2008.

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Vibration Absorbers for Cyclic Rotating Flexible Structures

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  1. Vibration Absorbers for Cyclic Rotating Flexible Structures Linear and Nonlinear Tuning Brian J. Olson Senior Staff Engineer The Johns Hopkins University Applied Physics Laboratory Steven W. Shaw Professor Michigan State University SMASIS08_5-7, Ellicott City, MD, October 28-30, 2008

  2. Agenda • Motivation • Review of Linear Tuning • Nonlinear Tuning • Design Recommendations • Conclusions & Future Work

  3. Motivation Bladed Disk Assemblies Sector Blade Shroud Pratt & Whitney JT9D-7J Engine Bladed Disk Assembly (Powers Boeing 747) Bladed Disk Assemblies Consist of a Cyclic Array of Rotating Substructures; 1 Blade + 1 Disk Portion = 1 Fundamental Sector

  4. Motivation Engine Order Excitation   Engine Order n Captures the nth Harmonic of Periodic Blade Excitation; Excitation Frequency is Proportional to Rotor Speed

  5. Motivation Engine Order Excitation Forward Traveling Wave (FTW) Engine Order Excitation Has a Discrete/Continuous Duality; Applied Loading is a BTW, FTW, or SW, Depending on n Relative to N

  6. Motivation Conditions For Resonance Possible Resonances Correspond to: ; Ideal Setting for Centrifugally-Driven, Order-Tuned Vibration Absorbers

  7. Motivation Order-Tuned Vibration Absorbers Absorber Primary System Restoring Force is Generated from Centrifugal Field due to Rotation; Order Tuning is Effective for Any Rotor Speed

  8. Motivation Vibration Reduction Via Order-Tuned Absorbers Can We Design to Attenuate Vibrations of ?

  9. Motivation Vibration Reduction Via Order-Tuned Absorbers Sleeves Tuned Dampers Chamber & End Caps Duffy et al 2000, 2001 Vibration Reduction Via Order-Tuned Absorbers is Not a New Idea; Previous Works Focus on Experimental Investigation

  10. Motivation Vibration Reduction Via Order-Tuned Absorbers First Systematic Analytical Treatment of Order-Tuned Absorbers Applied to a Cyclic Rotating Structure Under Engine Order Excitation

  11. Motivation Vibration Reduction Via Order-Tuned Absorbers Analytically Investigate Absorber Performance Using a Low-Fidelity, Lumped-Parameter, Perfectly Periodic Model

  12. Review of the Coupled Linear System Linear Absorber Tuning Order Linearize EOM and Impose Zero Blade Motions; Absorber Natural Freq. Proportional to Rotor Speed; is Fundamental Linear Design Parameter

  13. Review of the Coupled Linear System Effects of Detuning: The No-Resonance Zone Linear Absorber Tuning Only Mode n + 1 is Excited in the Steady-State; Entire Range of Absorber Designs for Which There are No Resonances

  14. Review of the Coupled Linear System Forced Response Absorbers Locked Linear Tuning Linear Overtuning Results in Resonance (Mode n + 1 = 4); Blades and Absorbers Respond In-Phase

  15. Review of the Coupled Linear System Forced Response Absorbers Locked Linear Tuning Sufficiently Large Undertuning Results in Resonance (Mode n + 1 = 4); Blades and Absorbers Respond Out-Of-Phase

  16. Review of the Coupled Linear System Forced Response Absorbers Locked Linear Tuning No Resonances For Linear Tuning Within The No-Resonance Zone; Valid For All Rotor Speeds; Robust to Parameter Uncertainty

  17. Mathematical Model Nonlinear Sector Blades and Absorbers are Dynamically Coupled Via m and Inter-Sector Coupling is Captured by n ; Absorber Path is Arbitrary

  18. Mathematical Model Absorber Path Softening Nonlinearity Hardening Nonlinearity Sets Linear Tuning (Prescribes Curvature at Path Vertex) Sets Nonlinear Tuning (Varies Curvature as increases)

  19. Formulation Scaled Sector Model Blade Dynamics (Linear) Absorber Dynamics (Weakly Nonlinear) Produces a Set of Models that are Amenable to Averaging; Choose Scaling s.t. Nonlinearity, Damping, & Coupling Appear at

  20. Formulation Averaged Sector Model Variation of Parameters Puts Scaled Equations Into a Form That Can Be Averaged; Captures the Desired TW Response Via

  21. Formulation Averaged Sector Model Detuning Introduced to Investigate Effects of Absorber Path Nonlinearity Near Primary Resonance and for Near-Ideal Linear Tuning

  22. Formulation Averaged Sector Model Nonlinear Frequency Response Approximated From Averaged Models; Captures Vibration Amplitudes and Captures Coupling Effects

  23. Features of the Forced Response Topology Key Consider Separately: • Isolated Nonlinear System Embodies the basic nonlinear features, except certain stability results • Coupled Nonlinear System Predicts instabilities of the desired traveling wave (TW) response Desired TW Frequency Response of (2) Qualitatively Corresponds to (1); Instabilities of TW Response Must Be Determined From (2)

  24. The Isolated Nonlinear System Frequency Response (h < 0) Linear Undertuning Sufficiently Large Linear Undertuning Results in Primary Resonance; Softening for (Shown) and Hardening for (Not Shown)

  25. The Isolated Nonlinear System Frequency Response (h < 0) Linear Undertuning Linear Overtuning Linear Overtuning Results in a Nonlinear Primary Resonance and an Isolated Nonlinear Auxiliary Resonance

  26. The Isolated Nonlinear System Frequency Response (h < 0) Linear Undertuning Slight Undertuning Linear Overtuning Primary Resonance Vanishes For Linear Tuning in No-Resonance Zone; Nonlinear Auxiliary Resonance Persists (Isolated From Lower Branch)

  27. The Coupled Nonlinear System Traveling Wave (TW) Response • Existence • Desired response is a TW of identical, constant-amplitude sector dynamics • Any other response type implies degraded absorber performance To Each Stationary Point Corresponds a Traveling Wave Response With The Same Stability Type

  28. The Coupled Nonlinear System Traveling Wave Response • Existence • Stability Jacobian Matrix is Block Circulant with 4 x 4 Blocks; Local Stability of Obtained From Block Decoupled Jacobian Matrices

  29. The Coupled Nonlinear System Possible Symmetry-Breaking Bifurcations • Identical to Jacobian matrix for isolated sector, except for small coupling-dependent resonance shift: • Predicts symmetry-preserving jump instabilities of the TW response • Predict symmetry-breaking instabilities of the TW response • None could be identified (based on extensive numerical studies) • Response of the coupled system is a TW • Local stability can be obtained from isolated sector No Symmetry-Breaking Instabilities of the TW Response; Local Stability of Coupled System Follows from Isolated Sector Model

  30. The Coupled Nonlinear System Frequency Response The Blades and Absorbers Respond in a Traveling Wave

  31. Conclusions Absorber Design Recommendations Feasible Absorber Design Corresponds to Slight Linear Undertuning in No-Resonance Zone; Absorber Paths Should Be Slightly Softening

  32. Conclusions • Summary and Key Findings • First systematic analytical study of its kind • Investigated effects of nonlinear absorber paths • No-resonance zone persists • Desired traveling wave response is stable • Absorber Design Recommendations • Select linear detuning within the no-resonance zone • 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 (effects of parameter/model uncertainty) • Experimental validation

  33. This work was supported by the National Science Foundation under grant CMS-0408866

  34. Collaborators BACKUP

  35. Background Engine Order Excitation Engine Order Excitation Has a Discrete/Continuous Duality; Applied Loading is a BTW, FTW, or SW, Depending on n Relative to N

  36. Background Engine Order Excitation Backward Traveling Wave (BTW)

  37. Background Engine Order Excitation Standing Wave (SW)

  38. Background Engine Order Excitation Forward Traveling Wave (FTW)

  39. Background Engine Order Excitation Standing Wave (SW)

  40. Linear Resonance Structure N = 10 Sectors, Blades/Absorbers Free Natural Frequency Rotor Speed Linear Absorber Tuning:

  41. Formulation Linear Resonance Structure of the Scaled System The Scaling Essentially Linearizes the Blade Dynamics While Capturing the Basic First-Order Effects of Absorber Path Nonlinearity

  42. Formulation Linear Resonance Structure of the Scaled System Linear Resonance Structure Qualitatively Persists Under the Scaling, Including the No-Resonance Zone

  43. Formulation Averaged Sector Model Detuning Introduced to Investigate Effects of Absorber Path Nonlinearity Near Primary Resonance and for Near-Ideal Linear Tuning

  44. Formulation Averaged Sector Model Nonlinear Frequency Response Approximated From Averaged Models; G Captures Vibration Amplitudes and g Captures Coupling Effects

  45. The Isolated Nonlinear System Critical Nonlinear Tuning • Solve for nonlinear tuning value for which blade amplitudes are zero • Depends on rotor speed and force amplitude • For proper linear undertuning requires undesirable hardening absorber path • Highly sensitive to parameter uncertainty Critical Nonlinear Tuning Depends on Rotor Speed and Force Amplitude; Nonlinearity Cannot Be Exploited to Improve Absorber Performance

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