INTRODUCTION - Signals, Systems, and Transforms - page 9

L

Mg

Figure 1.9

Simple pendulum.

rapidly is large. Hence, the differentiator amplifies any high-frequency noise in
However, some practical applications require the use of a differentiator. For
these applications, some type of high-frequency filtering is usually required before
the differentiation, to reduce high-frequency noise.

v i (t).

Simple Pendulum

We now consider a differential-equation model of the simple pendulum, which is
illustrated in Figure 1.9. The angle of the pendulum is denoted as the mass of the
pendulum bob is M , and the length of the (weightless) arm from the axis of rotation
to the center of the bob is L .
The force acting on the bob of the pendulum is then Mg , where g is the gravi-
tational acceleration, as shown in Figure 1.9. From physics we recall the equation of
motion of the simple pendulum:

u,

ML d 2 u(t)
dt 2

=-Mg sin u(t). (1.13)

This model is a second-order nonlinear differential equation; the term sin u(t) is non-

linear. (Superposition does not apply.)
We have great difficulty in solving nonlinear differential equations; however,
we can linearize (1.13). The power-series expansion for sin u is given (from

Appendix D) by

u 3
3! +

u 5

5! - Á .

sin u = u - (1.14)

For small, we can ignore all terms except the first one, resulting in when
is expressed in radians. The error in this approximation is less than 10 percent for
, is less than 1 percent for and
decreases as becomes smaller. We then express the model of the pendulum as,
from (1.13) and (1.14),

u sin u L u

u

u = 45° (p/4 radians) u = 14° (0.244 radians),

u

d 2 u(t)
dt 2

g
L u(t) = 0

+ (1.15)