The
differentiation theorem can be used to convert
differential
equations into
algebraic equations, which are easier to solve.
We will now show this by means of two examples.
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
forcedriven 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 Laplacetransform analysis goes for
all linear, timeinvariant (
LTI) systems. For
nonlinear and/or
timevarying systems, Laplacetransform 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 unitstep 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.
Consider now the
mass
spring oscillator depicted physically in
Fig.
D.3, and in
equivalentcircuit form in
Fig.
D.4.
Figure D.3:
An ideal mass sliding on a
frictionless surface, attached via an ideal spring to a rigid
wall. The spring is at rest when the mass is centered at .

Figure D.4:
Equivalent circuit for the massspring oscillator.

By
Newton's second law of motion, the
force applied to a mass
equals its mass times its acceleration:
By
Hooke's law for ideal springs, the compression force
applied to a spring is equal to the spring constant
times the
displacement :
By Newton's third law of motion (``every action produces an equal and
opposite reaction''), we have
. That is, the compression
force
applied by the mass to the spring is equal and opposite to
the accelerating force
exerted in the negative
direction by
the spring on the mass. In other words, the forces at the massspring
contactpoint sum to zero:
We have thus derived a secondorder
differential equation governing
the motion of the mass and spring. (Note that
in
Fig.
D.3 is both the position of the mass and compression
of the spring at time
.)
Taking the
Laplace transform of both sides of this
differential
equation gives
To simplify notation, denote the initial position and
velocity by
and
, respectively. Solving for
gives
denoting the modulus and angle of the
pole residue
, respectively.
From §
D.1, the inverse Laplace transform of
is
, where
is the
Heaviside unit step function at time 0.
Then by linearity, the solution for
the motion of the mass is
If the initial velocity is zero (
), the above formula
reduces to
and the mass simply oscillates sinusoidally at frequency
, starting from its initial position
.
If instead the initial position is
, we obtain
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