Math 240 Home
Launch Stability Lab
Poles of the Laplace Transform
and Behavior of Solutions
There are several situations where using Laplace transforms is easier than solving an equation using the techniques of chapter 2. One such situation is when you are more interested in certain qualitative features of the solution rather than the explicit formula for the solution. Some qualitative features of the solution can be read directly from the Laplace transform of the function. This means you will just have to carry out steps 1 and 2 of our paradigm to find the Laplace transform of the solution. Since taking the inverse Laplace transform in step 3 is the most complicated part of the paradigm, if we can skip this step, we can usually save some time and effort in answering our question.
The particular features that are easiest to read off the Laplace transform are whether the function tends to 0 or infinity as time tends to infinity. This is referred to as the stability of the solution. In addition, we can quickly tell whether the solution function is oscillating or not. We can also get a sense of how quickly the solution tends to 0 (if it does tend to 0).
It is often useful to design a system that decays quickly (think of shock
absorbers on a car) so this is a useful property to study.
- Launch the Stability lab (use the link at the top of the page). The
applet has two graph windows. The window on the right shows the graph of
the solution curve to the equation defined at the top of the applet. The
window on the left shows the poles of the Laplace transform of the
solution function. The Laplace tranform is a rational function, that is a
quotient of two polynomials. The poles (as you may remember from algebra)
are the zeros of the polynomial in the denominator of the Laplace
transform of the function. The poles are marked with an
the complex plane. If you get a double pole (a double root of the
polynomial in the denominator), then the
X will be
circled.The differential equation is initially set to
x'' + 9x
x'(0)=0. You can
check that the Laplace transform of the solution is
10s/(s2+9) and so the poles are the roots of
s2 + 9 = 0, which are
s = ±3i. By the way, it isn't possible to
adjust the scale of the two graph windows. Trying to add controls for that
just made things too crowded and you shouldn't need to adjust the scale
for anything we're doing in this lab.
- The initial equation,
x'' + 9x = 0, is
undamped. Increase the coefficient of
x' from 0 to 16 in
steps of 2. How do the poles of the Laplace transform change as you
increase the damping? How does the speed with which the solution converges
to 0 change as you increase the damping?
- Now experiment with changing the values of the different coefficients,
while leaving the forcing function set at 0. How can you distinguish
undamped, underdamped, critically damped, and overdamped equations from
the poles of the Laplace transform of the solution?
- Continue to experiment with changing the values of the coefficients of
x, only now pay
attention to the speed with which the solution tends to 0. How can you
judge how quickly the solution tends to 0 from the poles of the Laplace
- In general, changing the values of the initial conditions won't change
the poles in the left window. Can you explain why the initial values
usually don't affect the locations of the poles of the Laplace transform?
Can you find a second-order example (i.e. where the coefficient of
x'' is not 0) where changing the initial values will cause
just one of the poles to disappear? This question may be easier if you
start with a Laplace transform with just one pole and work backwards to
find the second-order initial value problem
- Now change the amplitude of the forcing function (the right-hand-side
of the equation) to something positive. This will add two more poles to
the left window. So far we have only considered equations where the
solution converges to 0, or at least stays bounded (in other words, we
have only considered stable equations). Our solution will tend to infinity
(and hence be unstable) if we have resonance. How can you distinguish
resonance from the poles of the Laplace transform?
- One other way to have solutions tending to infinity is to have a
negative amount of friction. This may seem ridiculous, but consider the
feedback loop when a microphone is placed in front of an amplifier. In
this applet, it is possible to enter a negative value for the coefficient
x'. Where are the poles of the Laplace transform when
the damping is negative?
Prepare a lab report for this lab which includes your answers to the
questions in bold face.
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©2002 Andrew G. Bennett