EE291E Hybrid Modeling of Microelectromechanical Systems Jason Clark BSAC UC Berkeley

What’s the Motivation Currently, there has been great success modeling the electrostatic and structural dynamics of MEMS REF: MEMCAD, Automm Automatic Generation of Dynamic Macromodels Boundary Element Analysis Finite Element Analysis

Motivation, cont. Comb-drive array Mirror Torsional hinge Modified Nodal Analysis MEMS modeling is complex due to multi-scales and multi- energy domains such as electro- magnetic, mechanical, thermal, digital/analog electrical, etc.

Motivation, cont. But a particular class of MEMS, such as actuators involving contact, has remained more of a challenge. Abe Lee : UC Berkeley, PhD dissertation (1992) Roger Hipwell : UC Berkeley, MS thesis (1998) Norman Tien : Cornell, thermal impact actuator (2000) *Richard Yeh : UC Berkeley, PhD dissertation (2001) Easier to make than to simulate in current FEA & MNA frameworks * test case

Progression in MEMS simulation CT : ODEs –E.g. MEMCAD, SUGAR CT+DE : Mixed signal systems –E.g. SUGAR CT+DE+FSM : Hybrid systems –E.g. Ptolemy Next step

Why has CT modeling of impact been impractical for MEMS simulation?

First test case: the Inchworm Motor Operation principles

Impacted State

Non-Impacted State

Non-impacted state Holder Mover M1 K1 F1 F2 FSM - A K2 M2 F1 = f(v1,v2) F2 = f(v3)

Impacted State Holder Mover M1 K1 F1 F2 FSM - B K2 M2

Hold State Holder M1 K1 F2 FSM - C Mover M1 K1 F1 K2 M2

Hybrid System CT+DT+FSM CT integration for FSM - A CT integration for FSM – B/C A B/C ti+1ti+2ti v1 v2 v3 DT: engage pull hold

Q - Switching Gap(X1) < 0.1  m F2 RXN < 1  N Gap(X2) < 0.1  m F2 RXN < 1  N & F1 RXN < 1  N Free Pull Hold Gap vs F RXN guards are used to tame Zeno behavior. They widen the marginal thresholds for switching. Inchworm Loop

Hybrid execution Time propagates to all components –t is global System is CT at each integration time point –Within DT and a FS FSM cannot make transitions during t+dt –Time stands still during transitions After dt FSM examines switching thresholds –At t = t + dt FSM makes transitions according to guards –Switching control FSM performs actions on the transition –Initial conditions are set for new state Zeno execution: An execution is Zeno is it contains an infinite number of transitions in a finte amount of time. An automaton is Zeno if it accepts a Zeno execution.

Sugar to Ptolemy Electrical Mechanical Coupled 1 st order ODE MNA matrix MNA solution vector Excitation vector where G=conductance ce=constitutional eqs C=capacitance c=C contribution V=voltage Q=charge M=mass D=damping K=stiffness F=force q=displacement & = Multi-domain state vector Input coupling matrix System matrix

Ptolemy Abstraction

E.g. Sticky Mass M1 M2 K1 K2 F1 FSM-A FSM-B/C