- How do we model a resonator when its forcing function is not continuous such as when a bell is struck?
The Laplace Transform
The laplace transform of a function only exists only if there exist numbers M and C such that: For all values of t > 0.
(Dirac Delta Function)
(Heviside Step Function)
|C f(t) (multiplication by a constant)||C F(s)|
|f(t-a)u(t-a), a>0 (time shift)|
|f(at), a>0 (time scale)|
The Laplace Transform is an integral transform. It is useful for solving differential equations with non-continuous forcing functions such as modeling the response of a resonator after applying some pulse. s can be a complex value.
Shifts in the time domain
Shifts in the frequency domain
Solving Differential Equations with Laplace Transforms
Transforming a differential equation into the s domain by means of the laplace transform makes its solution an algebraic problem. Then the solution must be transformed back into the t domain.