Note that the natural frequency of the swing is not influenced by the mass of the person in it. In other words' it makes no difference whether a swing has a large adult or a small child in it. It will have the about the same natural frequency. Slight differences can be caused by slightly different locations of the person's center of mass. This is located about two inches below the navel.
When people are sitting the center of
mass is in about the same place relative to the seat of the swing
regardless of whether the person is an adult or a child.
In this case even a small child can make
a large adult swing by pushing in sync with the swing's back and
forth cycle. The forcing function can also be provided by the person
in the swing. In this case the person in the swing shifts her center
of mass very slightly by changing the position of her legs or torso.
This creates a slight pushing force which makes the swing go higher
and higher. It takes a very small force but it has to be timed
After all, if it is a resonating system then it should be very dangerous to keep applying force in time with the swing's frequency. The answer is fairly simple. The equation given above is only good for small angles. When the swing goes beyond a certain height it is no longer possible for the person in it to apply the necessary small force in sync with the natural frequency because the natural frequency changes.
In other words the motion of the system
is naturally limited.
Visit a playground and measure the natural frequency of a swing. It should make very little difference whether the person in it is large or small. Attach a flimsy piece of thread to the person in the swing. Instruct them not to assist in making the swing move and then attempt to make the swing resonate by pulling on the thread without breaking it.
If the the force is applied in time with
the natural frequency of the swing it will make the swing resonate.