Interdisciplinary Applied Mathematics

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eovo



eo v2



(d + wo)2    (d + wo)3


Next, the elastic properties of the beam are taken into account. Employing energy methods (Shames and Dym, 1985) and excluding dynamic terms, the differential equation of motion is


L [w(x)j = —qe(x)+q(x),    (17.6)


where qe(x) and q(x) are transverse forces per unit length of electrical and mechanical origin, and L is the differential operator given by


L



d4    d2


dx4    dx2



(17.7)


where E and I are    the    Young’s    modulus    and    the    second    moment    of    iner


tia of the beam, respectively, and N is the applied axial force. Rewriting equation (17.6), we have


where



q(x)



be 0vq


(d + wo(x))2



v + Le [w(x)] ,



(17.8)



Le = L —



be qv2


(d + wo(x))3



L — ke(x).


The eigenvalues and the mode shapes (eigenvectors) of the governing equation (17.8) are now computed by solving a standard eigenvalue problem. Using the mode superposition principle, an infinite number of ordinary differential equations are formed (this topic is discussed in more detail in Section 17.4). Typically, a few mode shapes contain most of the mechanical energy, and these few modes can satisfactorily capture the mechanical deformation, thereby reducing the order of the problem.


In summary, we have a few ODEs to describe the parallel plate actuator, which can now be used to construct the equivalent circuit of the system in the same way as described earlier for the lumped parameter modeling. The advantages of the distributed parameter approach are the following: (i) It can be used to model continuous systems where most other methods fail. (ii) It can be incorporated into system simulators. (iii) Distributed parameter electrical devices can be coupled to the mechanical and the electrical terminal pairs as done in the lumped networks case. In a general case, the system has one electrical port characterized by the voltage v and the current i, and an infinite number of mechanical ports characterized by a generalized load and a generalized velocity. The disadvantages of the method are these: (i) It needs designer input, and test structures are required to verify whether the modeling results are correct. (ii) In most cases the conservative and dissipative energy domains have to be modeled separately. (iii) Since there is no unique representation possible, macromodel generation cannot be automated easily.

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