Figure
D.1 depicts a free mass driven by an external
force along
an ideal frictionless surface in one dimension. Figure
D.2
shows the
electrical equivalent circuit for this scenario in
which the external force is represented by a voltage source emitting
volts, and the mass is modeled by an
inductor
having the value
Henrys.

Figure D.1:
Physical diagram of an external force driving a mass
along a frictionless surface.
 |
Figure:
Electrical equivalent circuit of the
force-driven mass in Fig.D.1.
 |
From Newton's second law of motion ``

'', we have
Taking the unilateral
Laplace transform and applying the
differentiation theorem twice yields
Thus, given
Laplace transform of the driving force
,
initial mass position, and
-
initial mass velocity,
we can solve algebraically for

, the Laplace transform of the
mass position for all

. This Laplace transform can then be
inverted to obtain the mass position

for all

. This is
the general outline of how Laplace-transform analysis goes for
all linear, time-invariant (
LTI) systems. For
nonlinear and/or
time-varying systems, Laplace-transform analysis cannot, strictly
speaking, be used at all.
If the applied external force

is zero, then, by linearity of
the Laplace transform, so is

, and we readily obtain
Since

is the Laplace transform of the
Heaviside unit-step function
we find that the position of the mass

is given for all time by
Thus, for example, a nonzero initial position

and zero
initial velocity

results in

for all

; that
is, the mass ``just sits there''.
D.3 Similarly, any initial velocity

is integrated with
respect to time, meaning that the mass moves forever at the initial
velocity.
To summarize, this simple example illustrated use the Laplace
transform to solve for the motion of a simple physical system (an
ideal mass) in response to
initial conditions (no external driving
forces). The system was described by a
differential equation which
was converted to an algebraic equation by the Laplace transform.
Next Section: Mass-Spring Oscillator AnalysisPrevious Section: Differentiation