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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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