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Rotating Cyclic Systems with Order-Tuned Vibration Absorbers

Rotating Cyclic Systems with Order-Tuned Vibration Absorbers. Outline. Cyclic Structures Order-Tuned Absorbers Motivation & Background The Linear Problem The Nonlinear Problem Conclusions & Future Work. Relevant Previous Work. Order-Tuned Vibration Absorbers

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Rotating Cyclic Systems with Order-Tuned Vibration Absorbers

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  1. Rotating Cyclic Systems with Order-Tuned Vibration Absorbers

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

  3. 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, …

  4. Background Bladed Disk Assemblies

  5. Background Engine Order Excitation

  6. Background Order Excitation

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

  8. Order-Tuned Vibration Absorbers

  9. Order-Tuned Vibration Absorbers Torsional Vibration Reduction

  10. 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)

  11. Absorber Paths General Path Representation

  12. 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,

  13. Mathematical Model Equations of Motion

  14. Mathematical Model Equations of Motion

  15. Mathematical Model

  16. Cyclic Symmetry

  17. The Linearized System Sector Model

  18. The Linearized System System Model – M DOF/Sector

  19. The Linearized System System Model – M DOF/Sector

  20. Mathematical Preliminaries Circulant Matrices

  21. Mathematical Preliminaries Diagonalization of a Block Circulant

  22. Mathematical Preliminaries The Fourier Matrix

  23. Mathematical Preliminaries The Direct (Kronecker) Product

  24. Linear Vibration (Block) Decoupling the EOM

  25. Linear Free Vibration One DOF/Sector

  26. Linear Free Vibration One DOF/Sector

  27. Linear Free Vibration One DOF/Sector

  28. Linear Forced Vibration Steady-State Response

  29. Linear Forced Vibration Steady-State Physical Response

  30. Linear Forced Vibration Blade Response (Absorbers Locked)

  31. Linear Isolated Absorber Response Absorber Free, Blades Locked

  32. Linear Response N Blades with Absorbers

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

  34. Linear Response The Effects of Detuning, Strong Coupling

  35. Linear Response The Effects of Detuning

  36. Linear Response Frequency Response (zero damping)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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