How do you determine energy transferred to a trampoline when a ball is dropped on it?

Being physicists, we're going to start with a 1 dimensional trampoline, which is a mass $M$, on a spring with constant $k$, all damped ($c$), so that:

$$ Ma = F $$ $$ Mddot x = (- cdot x - k x) + F(t) $$

where $x=0$ is the equilibrium position, and $F(t)$ is an external driving force.

That can be rewritten as:

$$ ddot x(t) + 2 xi omega_0 dot x(t) + omega_0^2 x(t) = F(t)/M $$

where $omega_0 = sqrt{k/M}$ is the undamped angular frequency and $xi = frac c 2 frac 1 {sqrt{Mk}}$ the damping ratio.

Note that this is one of the most used equations in physics and engineering, so I'm not going to solve it here.

The question is now, how do you model the bounce?

I'd go with "a $pi$'s worth of $sin(omega_m t)$", or:

$$ F(t) = Asin(omega_m t) 0 le omega_m t le pildots $$

$0$ otherwise.

With

$$omega_m = sqrt{frac k {m+M}}$$

and

$$ frac 1 2 kA^2 = mgh $$

From here, you will get a frequency response function of the "trampoline" and then drive that with the frequency spectrum represented by $tilde F(omega)$, the Fourier transform of $F(t)$.