FYI I am an engineering student and would like to refrain from very complex notations. I am simply looking for a more accurate understanding of the laplace transform which is not an algebraic definition.
From my understanding for any function $f(t)$ we can express it via $\int(f(t)*e^{-st}))|[\infty,0]$. This provides some function $F(s)$ which is always a ratio of the zeros of F(s) by the poles of $F(s)$.
My main question is with any plotter online or simply plotting the $F(s)$ vs $s$, $s$ is noted as a one-dimensional axis but by definition, $s$ should be a combination of real and imaginary components $(\sigma + i*\omega)$. For example, if we have $F(s) = 1/(s^2+2s)$ and we were to plot $1/(x^2+2x)$ on Desmos would the x-values represent simply |real component| + |imaginary component|? or the square root sum of components? or simply the imaginary or real, etc.
I have also been curious about the idea that they always talk about the laplace transform moving a function from the time domain to the frequency domain. Why is this the case and how do we know this?
I have also been attempting to gain some intuition to understand how I can apply the laplace transform to a function via. a spring mass system analogy. I have been using this analogy since it seems to be popular and has applications between frequency and the $f(t)$ of the mass' motion. In these examples, we have an ODE representing the equation of motion and would usually solve $f(t)$ by guessing a solution in the form $e^{mt}$. There is some link when the laplace transform is done of the problem where the locations where $s>\infty$ is the same as the $m$ values of our solution. One thing to note is that the real part of $s$ should be analogous to the rate of exponential decay/growth of the mass oscillation. Whereas the imaginary part represents the natural frequency of oscillation.
- { The idea I have come up with is for the function being $f(t) = m*t'' +ct' +kt -$ Forced($t$). If we take the laplace transform of both sides, $f(t)$ can be represented as an additional forcing function of time. This would mean that our resulting function $F(s)$ could be viewed as the resulting (real and complex frequency) (s) and measure of amplitude ($y$-axis) of an additional force on the system needed for the equation to be true. }
-
Yeah honestly I thought I was getting somewhere with this but I'm really confused. If anyone has any intuition or a good analogy to explain what this is similar to it would be much appreciated.