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TABLE 8.3 Critical Load Coefficients
End Conditions
one end built-in, other free both ends built-in pin-joints at both ends
K coefficient 1/4 4 1

load, the beam will be unable to withstand the bending moment and will collapse. Consider the beam
in Fig. 8.5, subjected to load F with eccentricity e, resulting in lateral displacement of the tip δ. According
to the beam bending equation
2
∂ w
EI--------- = M = F d ++ w) (8.40)
(
e
∂x 2
where the boundary conditions are w(0) = 0, ∂w/∂x | x=0 = 0. The corresponding solution is
[
w = ( e + ) 1 – cos ( IE/Fx)] (8.41)
d
From w(L) = δ one has δ = e(1/coskL − 1), where k = IE/F . This solution looses stability when δ grows
out of bound, i.e., when coskL = 0, or kL = (2n + 1)π/2. From this condition the smallest critical load is

cr 2 2
F = p IE/4L (8.42)
The above analysis and Eq. (8.42) were developed by Euler. Similar conditions can be derived for other
types of beam supports. A general formula for the critical load can be written as
2
F cr = Kp IE/L 2 (8.43)
where several val ues of the coefficient K are given in Table 8.3.

8.5 Transducers

Transducers are devices capable of converting one type of energy into another. If the output energy is
mechanical work the transducer is called an actuator. The rest of the transducers are called sensors,
although in most cases, a mechanical transducer can also be a sensor and vice versa. For example the
capacitive transducer can be used as an actuator or position sensor. In this section the most common
actuators used in micromechatronics are reviewed.

Electrostatic Transducers

The electrostatic transducers fall into two main categories—parallel plate electrodes and interdigitated
comb electrodes. In applications where relatively large capacitance change or force is required, the parallel
plate configuration is preferred. Conversely, larger displacements with linear force/displacement charac-
teristics can be achieved with comb drives at the expense of reduced force. Parallel plate actuators are
used in electrostatic micro-switches as illustrated in Fig. 8.1. In this case the electrodes form a parallel
plate capacitor and the force is described by

2 2
Ae 0 e r V
F elec = ------------------------------------------- (8.44)
(
2 t 2 + e r d 0 – d)] 2
[
where A is the area of overlap between the two electrodes; t 2 is the thickness of insulating layer (silicon
dioxide, silicon nitride); l e is the length of fixed electrode; ε r is the relative permittivity of insulating layer;
V is the applied voltage; d 0 is the initial separation between the capacitor plates; and d is downward

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