Laplace Transforms 11: The Dirac Delta (Unit Impulse)
(Video 11 of more to come) In this video and the next, we explore another way to force a harmonic oscillator, such as a spring-mass system, by introducing the concept of an instantaneous force, like a sudden strike. (Quick note: the y on the vertical is not intended to represent the position function y--bad choice of notation on my part.)
This instantaneous force can be modeled using the unit impulse function, also known as the Dirac delta, a generalized function. We illustrate this idea by constructing a sequence of functions that approximate an impulse, explaining how this sequence impacts integration and ultimately leads to the Dirac delta function.
We create the form of the Dirac delta so that it is zero everywhere except at the point of impact. By using this approach, we model the effect of an instantaneous strike on a harmonic oscillator. We then demonstrate the Laplace transform of this Dirac delta function, showing that it leads to the same result whether using the geometric construction or an alternative characterization involving the Heaviside function.
In the next video, we will apply these concepts to solve differential equations with a Dirac delta forcing term, using the Laplace Transform.
Part 10: • Laplace Transforms 10: Examples with ...
Part 12: • Laplace Transforms 12: Dirac Delta Ex...
#differentialequations #mathematics #math #laplacetransformation #mathtutorial #engineeringmathematics #ordinarydifferentialequations #HeavisideFunction #UnitStepFunction #SignalProcessing #HarmonicOscillator
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