When we talk about damped oscillations, we're looking at how the size of the swings or movements gets smaller over time.

This happens because energy is lost. Things like friction or air pushing against the motion can slow it down.

We can describe this change using a simple formula:

A(t) = A₀ e^(-γt)

Here, A₀ is the starting swing size, and γ is the damping factor, which tells us how quickly the swings are getting smaller.

With damped oscillations, the frequency, or how often things swing back and forth, can also change a little. This smaller frequency is called the damped frequency. We can figure it out with another formula:

fᵈ = f₀ / √(1 - (r²))

In this formula, f₀ is the natural frequency, or how fast it would swing without any outside influence, and r is the damping ratio.

Now, let's talk about forced oscillations. In this case, something from outside is pushing or driving the system. If the frequency of this outside force matches the natural frequency of the system, it can keep the swing size the same.

When this special match happens, we call it resonance. During resonance, the size of the swings can grow very large. We can calculate the maximum swing size using another formula:

Aₘₐₓ = F₀ / (k - mω²)

In this formula, F₀ is the size of the driving force, k is the spring constant (which tells us how stiff the spring is), and ω is the frequency of the driving force.

So, to sum it up: damped oscillations get smaller over time due to energy loss, while forced oscillations can keep going strong if they are pushed correctly.