1. Rotating Cyclic Systems with Order-Tuned Vibration Absorbers

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

3. 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

4. Background Bladed Disk Assemblies

5. Background Engine Order Excitation

6. Order Excitation Background

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

8. Order-Tuned Vibration Absorbers

9. 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. Circulant Matrices Mathematical Preliminaries

21. Diagonalization of a Block Circulant Mathematical Preliminaries

22. 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 Strong Coupling Weak Coupling One DOF/Sector (Blades) – Weakly Nonlinear

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

51. Linear Blade & Nonlinear Absorber Weak Coupling

52. Linear Blade & Nonlinear Absorber

55. 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

56. 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