Rotating Cyclic Systems with Order-Tuned Vibration Absorbers.

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Presentation transcript:

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

Background Bladed Disk Assemblies

Background Engine Order Excitation

Order Excitation Background

Self-tuning Impact Damper Tuned Dampers Sleeves Chamber & End Caps Turbine Blade

Order-Tuned Vibration Absorbers

Torsional Vibration Reduction

History – Torsional Vibration Reduction Early designs –Sizing, physical arrangement –Linear tuning: local path curvature 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 Paths General Path Representation

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,

Mathematical Model Equations of Motion

Mathematical Model Equations of Motion

Mathematical Model

Cyclic Symmetry

The Linearized System Sector Model

The Linearized System System Model – M DOF/Sector

The Linearized System System Model – M DOF/Sector

Circulant Matrices Mathematical Preliminaries

Diagonalization of a Block Circulant Mathematical Preliminaries

The Fourier Matrix

Mathematical Preliminaries The Direct (Kronecker) Product

Linear Vibration (Block) Decoupling the EOM

Linear Free Vibration One DOF/Sector

Linear Free Vibration One DOF/Sector

Linear Free Vibration One DOF/Sector

Linear Forced Vibration Steady-State Response

Linear Forced Vibration Steady-State Physical Response

Linear Forced Vibration Blade Response (Absorbers Locked)

Linear Isolated Absorber Response Absorber Free, Blades Locked

Linear Response N Blades with Absorbers

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

Linear Response The Effects of Detuning, Strong Coupling

Linear Response The Effects of Detuning

Linear Response Frequency Response (zero damping)

Nonlinear Blade Response Strong Coupling Weak Coupling One DOF/Sector (Blades) – Weakly Nonlinear

Nonlinear Blade Response One DOF/Sector (Blades) – Strongly Coupled

Nonlinear Blade Response One DOF/Sector (Blades) – Strongly Coupled

Nonlinear Blade Response One DOF/Sector (Blades) – Strongly Coupled

Nonlinear Blade Response One DOF/Sector (Blades) – Strongly Coupled

Nonlinear Blade Response One DOF/Sector (Blades) – Weakly Coupled

Nonlinear Blade Response One DOF/Sector (Blades) – Weakly Coupled

Nonlinear Blade Response One DOF/Sector (Blades) – Weakly Coupled

Nonlinear Blade Response One DOF/Sector (Blades) – Weakly Coupled

Nonlinear Blade Response One DOF/Sector (Blades) – Weakly Coupled

Nonlinear Blade Response One DOF/Sector (Blades) – Weakly Nonlinear

Linear Blade & Nonlinear Absorber Assumptions and Scaling Goal: Capture nonlinear absorber behavior

Linear Blade & Nonlinear Absorber N Blade/Absorbers, Weak Coupling

Linear Blade & Nonlinear Absorber N Blade/Absorbers, Weak Coupling

Linear Blade & Nonlinear Absorber Weak 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