Rotating Cyclic Systems with Order-Tuned Vibration Absorbers.

Rotating Cyclic Systems with Order-Tuned Vibration Absorbers

Cyclic Structures

Order-Tuned Absorbers

• Motivation & Background

• The Linear Problem

• The Nonlinear Problem

• Conclusions & Future Work

Outline

• 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, …

Relevant Previous Work

BackgroundBladed Disk Assemblies

BackgroundEngine Order Excitation

Order ExcitationBackground

Background Self-tuning Impact Damper

Tuned Dampers

Sleeves

Chamber & End Caps

Turbine Blade

Order-Tuned Vibration Absorbers

Order-Tuned Vibration AbsorbersTorsional Vibration Reduction

History – Torsional Vibration Reduction

• Early designs– Sizing, physical arrangement– Linear tuning: local path curvature - 1930

• Implementations– Light aircraft engines, WWII– Helicopter rotors,1980’s– 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)

Absorber PathsGeneral Path Representation

Absorber PathsLinear 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,

Mathematical ModelEquations of Motion

Mathematical Model

Cyclic Symmetry

The Linearized SystemSector Model

The Linearized SystemSystem Model – M DOF/Sector

Circulant MatricesMathematical Preliminaries

Diagonalization of a Block CirculantMathematical Preliminaries

Mathematical PreliminariesThe Fourier Matrix

Mathematical PreliminariesThe Direct (Kronecker) Product

Linear Vibration(Block) Decoupling the EOM

Linear Free VibrationOne DOF/Sector

Linear Forced VibrationSteady-State Response

Linear Forced VibrationSteady-State Physical Response

Linear Forced VibrationBlade Response (Absorbers Locked)

Linear Isolated Absorber ResponseAbsorber Free, Blades Locked

Linear ResponseN Blades with Absorbers

Linear ResponseThe Effects of Detuning, Weak Coupling (like N=1)

Linear ResponseThe Effects of Detuning, Strong Coupling

Linear ResponseThe Effects of Detuning

Linear ResponseFrequency Response (zero damping)

Nonlinear Blade Response

Strong Coupling

Weak Coupling

One DOF/Sector (Blades) – Weakly Nonlinear

Nonlinear Blade ResponseOne DOF/Sector (Blades) – Strongly Coupled

Nonlinear Blade ResponseOne DOF/Sector (Blades) – Weakly Coupled

Nonlinear Blade ResponseOne DOF/Sector (Blades) – Weakly Nonlinear

Linear Blade & Nonlinear AbsorberAssumptions and Scaling

Goal: Capture nonlinear absorber behavior

Linear Blade & Nonlinear AbsorberN Blade/Absorbers, Weak Coupling

Linear Blade & Nonlinear AbsorberWeak Coupling

Linear Blade & Nonlinear Absorber

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

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

Rotating Cyclic Systems with Order-Tuned Vibration Absorbers.

Documents

Transcript of Rotating Cyclic Systems with Order-Tuned Vibration Absorbers.