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Unformatted text preview: Introduction to Laplace Transforms for Engineers C.T.J. Dodson, School of Mathematics, Manchester University 1 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. Definition 2 L [ f ( t )] = ˜ f ( s ) = Z ∞ e- st f ( t ) dt, for s > . (1.1) 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 en- counter 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 cal- culus 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. The amazing thing about using Laplace Transforms is that we can convert a whole ODE initial value problem into a Laplace transformed version as functions of s, 1 2 Introduction to Laplace Transforms simplify the algebra, find the transformed solution ˜ f ( s ) , then undo the transform to get back to the required solution f as a function of t. Interestingly, it turns out that the transform of a derivative of a function is a simple combination of the transform of the function and its initial value. So a calculus prob- lem is converted into an algebraic problem involving polynomial functions, which is easier....
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- Spring '09