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.
Physical diagram of an external force driving a mass
along a frictionless surface.
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
- 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
. This is
the general outline of how Laplace-transform analysis goes for
linear, time-invariant (LTI
) systems. For nonlinear
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
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
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
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 equivalent-circuit form in
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 .
Equivalent circuit for the mass-spring 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
By Newton's third law of motion (``every action produces an equal and
opposite reaction''), we have
. That is, the compression
applied by the mass to the spring is equal and opposite to
the accelerating force
exerted in the negative-
the spring on the mass. In other words, the forces at the mass-spring
contact-point sum to zero:
We have thus derived a second-order 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
To simplify notation, denote the initial position and velocity by
, respectively. Solving for
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
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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