Interdisciplinary Applied Mathematics

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Once all the mechanical systems are converted to electrical counterparts, a single representation of a system operating in more than one energy domain is finally obtained. Kirchhoffian conservation laws are then applied to solve the system. Commercial packages like SPICE28 (Quarles et al., 1987) can be    used    for    such    purposes.    This method    gives    a    better    understand

ing and visualization of the system and facilitates further analysis of the system in order to investigate the effects of the connecting subsystems or modifications to the system.

Lumped Parameter System

The lumped parameter approach is a reduced-order modeling method based on the equivalent circuit representation of microsystems. The basic idea in this method is to concentrate or lump the physical properties of the system, such as mass, stiffness, capacitance, and inductance, into single physical elements (Tilmans, 1996). The elements representing the mass are assumed

TABLE 17.1. Direct electromechanical analogies for lumped translational systems.    _

Mechanical Quantity

Electrical Quantity

Force: F

Voltage: V

Velocity: и

Current: %

Displacement: x

Charge: q

Mass: m

Inductance: L

Compliance: 1 jk

Capacitance: C

Viscous resistance: c

Resistance: R

to be perfectly rigid, and conversely, elastic elements have no mass. Lumped parameter modeling is typically valid as long as the wavelength of the signal is greater than all the dimensions of the device or the system under consideration. The device is finally represented as a set of lumped electrical network elements using the electrical-mechanical analogies. Once the circuit representation is constructed, commercially available circuit simulators such as SPICE can be used. Alternatively, dynamical state equations can be obtained from the network and simulated using standard math packages like MATLAB. There are two major issues in creating lumped-element macromodels: the first issue is to partition the continuum device into a network of lumped elements, and the second issue is to determine the parameter values for each element.

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