25-02-2013, 10:59 AM
Introduction to Laplace Transforms for Engineers
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What are Laplace Transforms, and Why?
This is much easier to state than to motivate! We state the definition in two ways,
first in words to explain it intuitively, then in symbols so that we can calculate
transforms.
Definition 1
Given f, a function of time, with value f(t) at time t, the Laplace transform of f is
denoted ˜ f and it gives an average value of f taken over all positive values of t such
that the value ˜ f(s) represents an average of f taken over all possible time intervals
of length s.
A short table of commonly encountered Laplace Transforms is given in Section 7.5.
Note that this definition involves integration of a product so it will involve frequent
use of integration by parts—see Appendix Section 7.1 for a reminder of the formula
and of the definition of an infinite integral like (1.1).
This immediately raises the question of why to use such a procedure. In fact the
reason is strongly motivated by real engineering problems. There, typically we encounter
models for the dynamics of phenomena which depend on rates of change of
functions, eg velocities and accelerations of particles or points on rigid bodies, which
prompts the use of ordinary differential equations (ODEs). We can use ordinary calculus
to solve ODEs, provided that the functions are nicely behaved—which means
continuous and with continuous derivatives. Unfortunately, there is much interest in
engineering dynamical problems involving functions that input step change or spike
impulses to systems—playing pool is one example. Now, there is an easy way to
smooth out discontinuities in functions of time: simply take an average value over
all time. But an ordinary average will replace the function by a constant, so we use
a kind of moving average which takes continuous averages over all possible intervals
of t. This very neatly deals with the discontinuities by encoding them as a smooth
function of interval length s
Solving ODEs and ODE Systems
The application of Laplace Transform methods is particularly effective for linear
ODEs with constant coefficients, and for systems of such ODEs. To transform an
ODE, we need the appropriate initial values of the function involved and initial
values of its derivatives. We illustrate the methods with the following programmed
Exercises.
Impulse problems
Laplace transform methods are particularly valuable in handling differential equations
involving impulse and step functions. The problem in the Exercise below
represents the dynamics of a point, initially at rest, moving away from the origin
along the y-axis under a constant acceleration of value 10 for 0 t < 1 and an extra
impulse acceleration of size 10 is applied at t = 1. This is like a simple rocket boost,
but can you solve it any other way? We use the Dirac impulse function (t − a)
which is nonzero at t = a,