Interdisciplinary Applied Mathematics

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The advantages of the lumped parameter method are as follows: (i) It is easy to use and can be easily incorporated into system simulators. (ii) Equivalent lumped resistors treated external to the system can be used to model dissipation. (iii) The equivalent circuit representation can be used to analyze complex structures and coupled subsystems with several electrical and mechanical ports. The disadvantages of the method are these: (i) Unlike pure electric circuits, mechanical structures do not offer a clean

FIGURE 17.5. Continuously distributed system with infinite degrees of freedom.


mapping between the geometry and the corresponding network analogy. (ii) No CAD tools are currently available that can automatically construct an energetically correct lumped-element topology directly for an arbitrary device geometry. (iii) Large-signal and nonlinear analysis is cumbersome, difficult, and error-prone. (iv) In most cases the conservative and dissipative energy domains are to be modeled separately.


Distributed Parameter System


In a distributed parameter system (see (Tilmans, 1997)), the mass, compliance, capacitance, etc., are not easily identifiable as lumped elements at individual points. These elements are, instead, continuously distributed throughout the system. Figure 17.5 shows another parallel plate actuator, but in this case, in contrast to the lumped parameter case, the mass is a continuously deformable beam with a uniformly distributed load on it. The electrical and mechanical domains are coupled either through the boundary of    the    flexible    beam    or    throughout    the    entire    system as in    the    case    of


transducers employing piezoelectric materials. In such cases, it is difficult to distinguish between the mechanical and electrical forces, and the lumped parameter system cannot be easily used to extract the circuit parameters. Instead, the distributed parameter approach needs to be employed. The fundamental difference between lumped parameter models and distributed parameter systems is that while the former method has a finite number of degrees of freedom, the latter has an infinite number of degrees of freedom. The lumped parameter models and the distributed parameter models are just two distinct mathematical models of the same physical system and the distributed parameter approach can be considered as a more general approach compared to the lumped parameter approach (Tilmans, 1997).

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