Vibration Absorbers for Cyclic Rotating Flexible Structures Brian J. Olson Linear and Nonlinear Tuning 9 October 2008 A1F Lunchtime Seminar

slide 2 Collaborators  Steve Shaw Professor Department of Mechanical Engineering Michigan State University  Christophe Pierre Dean College of Engineering McGill University

slide 3 MOTIVATION & BACKGROUND PART 1

slide 4 Motivation Bladed Disk Assemblies Bladed Disk Assemblies Consist of a Cyclic Array of Rotating Substructures; 1 Blade + 1 Disk Portion = 1 Fundamental Sector Sector Pratt & Whitney JT9D-7J EngineBladed Disk Assembly Blade Shroud (Powers Boeing 747)

slide 5 Engine Order n Captures the n th Harmonic of Periodic Blade Excitation; Excitation Frequency is Proportional to Rotor Speed Motivation Engine Order Excitation

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

slide 7 Can We Design to Attenuate Vibrations of ? Motivation Vibration Reduction Via Order-Tuned Absorbers

slide 8 Motivation Vibration Reduction Via Order-Tuned Absorbers 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 Tuned Dampers Sleeves

slide 9 Motivation Vibration Reduction Via Order-Tuned Absorbers First Systematic Analytical Treatment of Order-Tuned Absorbers Applied to a Cyclic Rotating System Under Engine Order Excitation

slide 10 Motivation Vibration Reduction Via Order-Tuned Absorbers Analytically Investigate Absorber Performance Using a Perfectly Periodic, Low-Fidelity (Lumped-Parameter) Model

slide 11 1.Quantify and Understand the Underlying Linear Resonance Structure; 2.Design Absorbers to Attenuate Blade Vibrations Near Resonance; and 3.Generalize to Include Effects of Nonlinear Absorber Paths Motivation Goals of This Effort 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?

slide 12 Agenda 1.Background Frequency-Tuned vs. Order-Tuned Vibration Absorbers Cyclic Systems Theory of Circulants / Mathematical Preliminaries Nonlinear Vibration 101 Engine Order Excitation 2.Linear Absorber Tuning Model / Formulation Modal Analysis / Forced Response Linear Resonance Structure / Absorber Tuning Effects of Damping 3.Nonlinear Absorber Tuning Mathematical Model / Path Selection Formulation: Scaling / Averaging Traveling Wave Forced Response / Stability Nonlinear Absorber Tuning 4.Conclusions Absorber Design Recommendations Summary of Contributions Directions for Future Work

slide 13 Background Frequency-Tuned Vibration Absorbers Primary System 1-DOF Oscillator Represents One Structural Mode of Primary System; How Can Vibration Amplitudes Be Attenuated Near Resonance?

slide 14 Background Frequency-Tuned Vibration Absorbers Attach Auxiliary Substructure and Tune It’s Nat. Freq. to Excitation Freq. Tuning is Effective at (or Near) a Fixed Excitation Frequency Primary System Vibration Absorber

slide 15 Background Order-Tuned Vibration Absorbers Restoring Force is Generated from Centrifugal Field due to Rotation; Order Tuning is Effective for Any Rotor Speed Primary System Absorber

slide 16 Background Linear Systems Feature Unique, Globally Stable Amplitudes of Vibration Linear vs. Nonlinear Frequency Response Linear Spring

slide 17 Background Nonlinearity Bends Frequency Response Curve; Results in Multi-Valued Amplitudes Linear vs. Nonlinear Frequency Response Linear Spring Bend Nonlinear Spring

slide 18 Background Softening vs. Hardening Response Softening Spring Hardening Spring Softening Nonlinearity Bends Resonance to the Left; Hardening Nonlinearity Bends Resonance to the Right Bend Softening Response Hardening Response

slide 19 Background Some Nonlinear Branches Are Unstable; Stability and Bifurcation Softening Spring Hardening Spring Stable Unstable

slide 20 Background Stability and Bifurcation Softening Spring Hardening Spring Stable Unstable Bifurcation Some Nonlinear Branches Are Unstable; Results in Jump Bifurcations as Excitation Frequency is Increased

slide 21 Background Linear vs. Nonlinear Frequency Response Softening Spring Hardening Spring Bifurcation Stable Unstable Some Nonlinear Branches Are Unstable; Results in Jump Bifurcations as Excitation Frequency is Decreased

slide 22 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

slide 23 Background Engine Order Excitation Backward Traveling Wave (BTW)

slide 24 Background Engine Order Excitation Standing Wave (SW)

slide 25 Background Engine Order Excitation Forward Traveling Wave (FTW)

slide 26 Background Engine Order Excitation Standing Wave (SW)

slide 27 LINEAR ANALYSIS PART 2 Model / Formulation Modal Analysis / Forced Response Linear Resonance Structure / Absorber Tuning Effects of Damping Summary Mathematical Model and the

slide 28 Mathematical Model Bladed Disk Assembly With Absorbers Engine Order Excitation

slide 29 Mathematical Model Linearized System Model EOM Derived From Lagrange’s Formulation, Then Linearized; System Matrices Are Block Circulant with 2 x 2 Blocks

slide 30 Modal Analysis Block Decoupling The Equations of Motion 2N-DOF System is Block-Decoupled Via a Standard Unitary Transformation to N, 2-DOF Systems in Modal Space

slide 31 Modal Analysis Steady-State Modal Response

slide 32 Only Mode p = n + 1 is Excited in the Steady-State Modal Analysis Steady-State Modal Response

slide 33 Steady-State (SS) Response of Coupled 2N-DOF System is Obtained From the Solution of a Single Harmonically-Forced 2-DOF System Modal Analysis Steady-State Modal Response

slide 34 Special Cases 1.Blades Locked, Absorbers Free 2.Blades Free, Absorbers Locked 3.Single Isolated Sector, Blade/Abs Free Gives Linear Absorber Tuning Order Benchmarks for Absorber Performance Demonstrates the Essential Features of the Coupled System Topology Key

slide 35 Special Case (1) Blades Locked, Absorbers Free Case (1) Motivates Linear Absorber Tuning Order , , and  are Design Parameters Linear Absorber Tuning Order Natural Frequency Rotor Speed

slide 36 Special Case (2) Blades Free, Absorbers Locked Case (2) Provides a Benchmark to Assess Absorber Performance Natural Frequency Rotor Speed Natural Frequency Rotor Speed Campbell DiagramFrequency Response

slide 37 Special Case (2) Blades Free, Absorbers Locked Case (2) Provides a Benchmark to Assess Absorber Performance Mode n + 1 = 4 is Excited Natural Frequency Rotor Speed Amplitude Rotor Speed Campbell DiagramFrequency Response

slide 38 Special Case (3) Single Isolated Sector, Blade/Absorber Free Case (3) Demonstrates Essential Features of the Full Coupled System Natural Frequency Rotor Speed

slide 39 Special Case (3) Single Isolated Sector, Blade/Absorber Free Case (3) Demonstrates Essential Features of the Full Coupled System Rotor Speed Natural Frequency

slide 40 Special Case (3) Single Isolated Sector, Blade/Absorber Free Case (3) Demonstrates Essential Features of the Full Coupled System Rotor Speed Natural Frequency

slide 41 Special Case (3) Single Isolated Sector, Blade/Absorber Free Case (3) Demonstrates Essential Features of the Full Coupled System Rotor Speed Natural Frequency

slide 42 Special Case (3) Single Isolated Sector, Blade/Absorber Free Ideal Absorber Tuning: Rotor Speed Natural Frequency

slide 43 Special Case (3) Single Isolated Sector, Blade/Absorber Free Linear Absorber Tuning: Rotor Speed Natural Frequency

slide 44 Linear Resonance Structure N = 10 Sectors, Blades/Absorbers Free Rotor Speed Natural Frequency Linear Absorber Tuning:

slide 45 Linear Resonance Structure Effects of Detuning: The No-Resonance Zone Linear Tuning

slide 46 Linear Forced Response Linear Tuning Absorbers Locked e.o. Line

slide 47 Linear Forced Response Linear Tuning Absorbers Locked e.o. Line

slide 48 Linear Forced Response Linear Tuning Absorbers Locked e.o. Line

slide 49 Absorbers Free Linear Forced Response Blade Motions are Completely Eliminated for Ideal Absorber Tuning; The Absorbers Respond in a TW Response to the TW Excitation

slide 50 Absorbers Free Linear Forced Response Some Residual Blade Motions For Tuning Within No-Resonance Zone; Offers Good Robustness to Parameter and Model Uncertainty

slide 51  Investigated linearized, perfectly periodic, lumped- parameter, bladed disk model fitted with order-tuned absorbers  The absorbers are effective  No-resonance zone  Ideal tuning results in complete reduction of blade motions  May be destroyed in the presence of model/parameter perturbations  Slight undertuning results in good reduction of blade motions, robustness to parameter uncertainty, and no resonances over full range of rotor speeds Linear Absorber Design Summary Absorbers Should Be Tuned Within The No-Resonance Zone

slide 52 NONLINEAR ANALYSIS PART 3 Mathematical Model / Path Selection Formulation: Scaling / Averaging TW Forced Response / Stability NL Absorber Tuning Summary

slide 53 Mathematical Model Nonlinear Sector Blades and Absorbers are Dynamically Coupled via  and Inter-Sector Coupling is Captured by  ; Absorber Path is Arbitrary

slide 54 Mathematical Model Absorber Path Sets Linear Tuning (Prescribes Curvature at Path Vertex) Sets Nonlinear Tuning (Varies Curvature as increases) Softening Nonlinearity Hardening Nonlinearity

slide 55 Formulation Scaled Sector Model Produces a Set of Models that are Amenable to Averaging; Choose Scaling s.t. Nonlinearity, Damping, & Coupling Appear at Blade Dynamics (Linear) Absorber Dynamics (Weakly Nonlinear)

slide 56 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

slide 57 Formulation Linear Resonance Structure of the Scaled System Linear Resonance Structure Qualitatively Persists Under the Scaling, Including the No-Resonance Zone

slide 58 Formulation Averaged Sector Model Variation of Parameters Puts Scaled Equations Into a Form That Can Be Averaged; Captures the Desired TW Response

slide 59 Formulation Averaged Sector Model Detuning Introduced to Investigate Effects of Absorber Path Nonlinearity Near Primary Resonance and for Near-Ideal Linear Tuning

slide 60 Formulation Averaged Sector Model Nonlinear Frequency Response Approximated From Averaged Models; G Captures Vibration Amplitudes and g Captures Coupling Effects

slide 61 Features of the Forced Response Consider Separately: 1.Isolated Nonlinear System Embodies the basic nonlinear features, except certain stability results Predicts instabilities of the desired TW response Topology Key 2.Coupled Nonlinear System Freq. Resp. and Local Stability of (2) Qualitatively Corresponds to (1); Instabilities of the Desired TW Must Be Determined From (2)

slide 62 The Isolated Nonlinear System Frequency Response (  < 0) e.o. Line

slide 63 The Isolated Nonlinear System e.o. Line Frequency Response (  < 0)

slide 64 The Isolated Nonlinear System e.o. Line Frequency Response (  < 0)

slide 65 The Isolated Nonlinear System Critical Nonlinear Tuning Nonlinearity Cannot Be Exploited to Improve Absorber Performance  Solve for nonlinear tuning value for which blade amplitudes are zero ( ū = 0 )  Highly sensitive to parameter uncertainty  Depends on rotor speed and force amplitude  For proper linear undertuning ( < 0 ) requires undesirable hardening absorber path

slide 66 The Coupled Nonlinear System Traveling Wave Response

slide 67 The Coupled Nonlinear System Possible Symmetry-Breaking Bifurcations

slide 68 The Coupled Nonlinear System Possible Symmetry-Breaking Bifurcations The Nonlinear Response is a Traveling Wave

slide 69 The Coupled Nonlinear System Frequency Response

slide 70 Summary Nonlinear Absorber Design  No-resonance zone qualitatively persists  Nonlinearity cannot be exploited to improve performance  No instabilities of the desired TW response

slide 71 CONCLUSIONS PART 4 Recommendations for Absorber Design Summary of Contributions Directions for Future Work

slide 72 Conclusions Absorber Design Recommendations

slide 73 Conclusions  Summary and Key Findings  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

slide 74 This work was supported by the National Science Foundation under grant CMS