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

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

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

#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

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

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

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

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

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

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

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

Experiment: 2DOF TMD for a free-free beam

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)

Passive Vehicle Suspension

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:

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)

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

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  m Ideal output of “skyhook” system red--without control blue--with control ( 1.23Hz: one natural freq of the 2 nd stage )