Interdisciplinary Applied Mathematics

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FIGURE 17.6. Element stamp for a resistor.


FIGURE 17.7. Schematic diagram of the parallel plate actuator with damping.


case, the Kirchhoff’s circuit laws are replaced by the Newtonian equations of motion. The equations governing the dynamics of rigid bodies are given by


»У’.Рк =0 and ^2 Tk =


к    к


where F and T stand for forces and torques, respectively. The similarity between Kirchhoffian current laws and these equations is apparent. However, one difference is that since microsystems involve both mechanical and electrical domains, there are both mechanical and electrical variables and equations contained in element stamps for microsystems. We illustrate the development of an element stamp for the parallel plate actuator shown in Figure 17.2. External (air) damping is also considered and is represented by the external damper (c) as shown in the schematic diagram in Figure 17.7. The governing equations for the system are given by


d2x dx    e0A    V2


ae+ca= F =    [(SWyr



and i



dq


dt



д Ге0AV


dt d + x


Trapezoidal integration and linearization of the governing equations gives


rise to the following simple algebraic equation:


dFe    , dFe


-K~Vt + ~7—Vb


dvt    dvb



0,


FIGURE 17.8. Element stamp for the parallel plate actuator.


h    h    h


muT H cu T + k—x H—


2    2    2


where ux = dx/dt, h is the time step of integration, and vt and vb are the voltages on the beam and the ground element, respectively. Using the nodewise analysis as described above for the resistor, the element stamp for this MEM actuator is shown in Figure 17.8. The stamp has four rows, two for the electrical equations and two for the mechanical-dynamical equations. The four columns correspond to the four variables, namely, vt, vb, x, and uxThis method takes much less time to simulate in a circuit simulator. It provides a compact and efficient way of adding the contribution of a particular element to the overall system. Since built-in models are used, this procedure can handle only those devices that can be described by the built-in models. This might not be a disadvantag, since all MEMS devices can be described in terms    of    a    set    of basic    elements.    The    computational    effort    required is

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