1. Decentralized Control Applied to Multi-DOF Tuned-Mass Damper Design Decentralized H2 Control Decentralized H  Control Decentralized Pole Shifting Decentralized H2 with Regional Pole Placement Lei Zuo and Samir Nayfeh

2. Control View of SDOF TMD Spring: feedback relative displacement with gain k 2 Damper: feedback relative velocity with gain c 2 k2k2 c2c2 u u = k 2 (x 2 - x 1 )+c 2 ( ) SDOF TMD  MDOF TMD: ---- To make use of other degree of freedoms ---- Better vibration suppression ---- To damp multiple modes with one mass damper

3. Formulation for MDOF TMD Systems The mass-spring-damper systems can be cast as a Decentralized Static Output Feedback problem Cost Output Measurement Disturbance 0 0

4. #PerformanceDisturbanceApproach 1Decentralized H 2 /LQr.m.s. response (impulse energy) white noisegradient-based 2Decentralized H  peak in frequency domain worst-case sinusoid LMI iteration/ gradient-based 3Pole shiftingmodal dampingunknown  -subgradient 4Decentralized H 2 + pole constraint r.m.s. response +transient char. partially-known white noise Methods of multipliers decentralized control for different disturbances and performance requirements

5. k1k1 k2k2 c1c1 c2c2 2d Minimal ||H|| 2 of x 0  x s versus  /d Minimal ||H|| 2 Radius of gyration / location (  /d) mass ratio m d /m s =5% 2DOF TMD for Single Mode Vibration   /d=1: two separate SDOF TMDs   /d=  : traditional SDOF TMD   /d=1/ : 2DOF TMDs (uniform)   /d=0.780: 2DOF TMDs (optimal)     k1k1 c2c2 k1k1 c2c2

6. 2DOF TMD: Decentralized H    /d=  Normalized Frequency Magnitude x s /x 0   /d=1   /d=1/sqrt(3)   /d=0.751 k1k1 k2k2 c1c1 c2c2 2d mass ratio m d /m s =5% 2DOF TMD can be better than the traditional SDOF TMD and two separate TMDs

7. 2DOF TMD: Negative Damping mass ratio m d /m s =5%,  /d=0.2 Much better performance if one of the damper can be negative. A new reaction mass actuator

8. Application: Beam Splitter of Lithography Machine flexures beam splitter (mockup) table (Acknowledgement: Thanks to Justin Verdirame for making this mockup)

9. 6DOF TMD for 6DOF Beam Splitter excitation accelerometer spring-dashpot connections mass damper

10. Measured T.F. of 6DOF TMD SIX modes are damped well just by using ONE secondary mass Phase (deg) Magnitude(dB)

11. Decentralized Pole Shifting 2DOF TMD for a free-free beam, 72.7" long Objective: To maximize the minimal damping of some modes Method: nonsmooth, Minimax (subgradient + eigenvalue sensitivity) cup plungerblade adjustable screw

12. Experiment: 2DOF TMD for a free-free beam

13. Vibration Isolation/Suspension Passive Vehicle Suspension: Decentralized H2 optimization 6DOF Active Isolation: Modal Control (collaborated with MIT/Catech LIGO) Dynamic Sliding Control for Active Isolation (with Prof Slotine)

14. Passive Vehicle Suspension

16. Sliding Control for Frequency-Domain Performance Conventional Sliding Surface Frequency-Shaped Sliding Surface We can design L i (s) to meet the frequency performance requirement Control force Disturbance force Coupling due to non-proportional damping In mode space:

17. Physical Interpretation of the Frequency-Shaped Sliding Surface Magnitude (dB) a 0 =2  (0.1  2  )  0.7 b 0 =(0.1  2  ) 2 For another case sky Take b 1 =0, on the sliding surface Skyhook ! Frequency (Hz)

18. Case Study: 2DOF Isolation M 1 =500 kg, I 1 =250 kg m 2, l 1 =1.0m, l 2 =1.4 m,  1 =5.42 Hz,  1 =1.01%  2 =9.56 Hz,  2 =1.41% l1l1 l2l2 Magnitude (dB) target x 1 /x 0 x 2 /x 0

19. Simulation Results Ground x 0 =0.01sin(1.23  2  t) meter X 1 (m) X 2 (m) Ideal Output (m) X 1 (m) 6.6  10 -5 m Ideal output of “skyhook” system red--without control blue--with control ( 1.23Hz: one natural freq of the 2 nd stage )