When only discrete elements are used to model a physical system, the associ-
ated system of governing equations is referred to as a discrete system or a
lumped-parameter system. In these cases, as will become evident in later chap-
ters, since a finite number of independent displacement or rotation coordinates
suffice to describe the position of a physical system, discrete systems are also
referred to as finite degree-of-freedom systems. When a distributed element is
used to model a physical system, the associated system of governing equations
is referred to as a distributed-parameter system or a continuous system. In this
case, one or more displacement functions are needed to describe the position of
a physical system. Since a function is equivalent, in a sense, to specifying the
displacement at every point of the physical system or the displacements at an
infinite number of points, distributed-parameter systems are also referred to as
infinite degree-of-freedom systems.
2.5.2 A Microelectromechanical System
In Figure 2.24, a microelectromechanical accelerometer
12
is shown along with
the vibratory model of this sensor. In this sensor, the dimensions of the can-
tilevered structure are of the order of micrometers and the weight of the end
mass is of the order of micrograms. A coating on top of the structure serves as
one of the faces of a capacitor and another layer below the structure serves as
another face of the capacitor. The gap between the capacitor faces changes in
response to the accelerations experienced by the sensor, and the change in volt-
age across this capacitor is sensed to determine the acceleration.
In constructing the vibratory model, the inertia of the cantilevered struc-
ture is ignored and this structure is represented by an equivalent spring with
stiffness k. The mass of the cantilevered structure is assumed to be negligible
and the end mass is modeled as a point mass of mass m. Consequently, the mo-
tion of this inertial element is described by a single generalized coordinate x,
and the model is an example of a single degree-of-freedom system. The elec-
trostatic force due to the capacitor acts directly on the mass, while the acceler-
ation to be measured acts at the base of the vibratory model. The electrostatic
force that acts directly on the inertial element is an example of a direct excita-
tion, while the acceleration acting at the base is an example of a base excita-
tion. In a refined model of the system, the mass of the cantilevered structure
can also be lumped together with that of the end mass to obtain an effective
point mass. No damping elements are used in constructing the vibratory model
because the physical system has “very low” damping levels.
Single degree-of-freedom systems are treated at length in Chapters 3
to 6. In particular, the response of a single degree-of-freedom system subjected
to a base excitation or direct excitation such as that shown in Figure 2.24 is dis-
cussed in Section 5.5.
y
$
2.5 Model Construction 55
12
K. E. Petersen, A. Shartel, and N. F. Raley, “Micromechanical accelerometer integrated with
MOS detection circuitry,” IEEE Transactions of Electronic Devices, Vol. ED-29, No. 1,
pp. 23–27 (1982).