1. Nonlinear dynamics of a rotor contacting an elastically suspended stator 1 st International Conference on Vibro-Impact Systems Loughborough, UK, July 20-22, 2006 S. Popprath* and H. Ecker Institute of Mechanics and Mechatronics Vienna University of Technology, Austria

2. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 2 Overview Introduction and Motivation Mathematical Model Numerical solution method Results Conclusions

3. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 3 Introduction and Motivation Vibratory system

4. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 4 Introduction and Motivation Vibro-impact system

5. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 5 Introduction and Motivation Actual background – vertical rotor test rig Stator Disk

6. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 6 Mathematical model Rotor and elastically suspended stator 4 DOF system Rotor: x r,y r Stator: x s,y s  = const 2 Imperfections Rotor unbalance Center offset

7. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 7 Mathematical model Rotor and elastically suspended stator Contact system

8. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 8 Mathematical model Equations of motion RotorStator Radial intrusion depth

9. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 9 Mathematical model CoordinatesContact forces vtvt d y off x off r rs

10. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 10 Mathematical model Dimensionless system equations Rotor Stator

11. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 11 Mathematical model Mass ratio Stiffness ratio Physical damping ratio Ratio of damping ratios Dimensionless rotor speed

12. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 12 Numerical solution method Numerically stiff problem Gear‘s algorithm Detection of state events (contacts) Start of contact phase End of contact phase Force condition D>0 Geometric condition

13. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 13 Results – Rotor and Stator Orbits Results Rotor orbit and stator motion for a 2p/2c-Orbit rotor stator 2p/2c-Orbit … 2 periods and 2 contacts

14. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 14 Results – Bifurcation Diagram for  Results M = 0.01 K = 3 C = 66.667 Parameters Z = 384.9 Last 100 points Last 1000 points Lossless Contact C h =0,  =0

15. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 15 Results – Rotor Orbits and Poincare-Maps Results  = 0.82 M = 0.01 K = 3 C = 66.667 Parameters Z = 384.9  = 0.774 M = 0.01 K = 3 C = 66.667 Parameters Z = 384.9

16. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 16 Results – Bifurcation Diagram for M Results M = 10 -2 ÷10 3 K = 3 C = 66.6 Parameters Z = 384.9 ÷ 1.217 Last 100 points Last 1000 points Lossless Contact C h =0,  =0  = 0.799

17. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 17 Results - Bifurcation Diagram for M Results M = 10÷100 K = 3 C = 12.550 ÷ 39.686 Parameters Z = 2.291 Lossless Contact C h =0,  =0 Last 100 points Last 1000 points  = 0.799

18. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 18 M = 10÷100 K = 3 C = 25.100 ÷ 79.373 Parameters Z = 4.583 Results - Bifurcation Diagram for M Results Last 100 points Last 1000 points Lossless Contact C h =0,  =0  = 0.799

19. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 19 Results - Bifurcation Diagram for M Results M = 10÷100 K = 3 C = 41.833 ÷ 132.288 Parameters Z = 7.638 Last 100 points Last 1000 points Lossless Contact C h =0,  =0  = 0.799

20. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 20 Conclusion Flexible rotor contacting an elastically suspended stator was investigated System exhibits rich dynamic behavior (periodic, quasi-periodic and chaotic solutions) Damping ratio has a large influence on the occurance of periodic and non-periodic solutions Still a basic system but already high dimension of the parameter space

21. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 21 Thanks for Your Attention Mass ratio M Deflection X r Rotor speed 

22. Measurement and Actuator Technology Introduction and Motivation Mathematical model Results Conclusion Numerical solution method S. Popprath and H. Ecker1st International Conference on Vibro-Impact Systems 22 Vertical Rotor Test Rig - Results Simulation results Measurements