Dynamics of swinging My husband built a 20'-high tree swing, using 3/16"-welded-link chains to suspend it approximately 2' off of the ground. To our disappointment, pumping barely gets the swing going, with little height elevation. Even after being pushed to attain a decent height/arc (for a good swinging experience), pumping will not then maintain the height/arc. Is this a factor of having used heavy chain versus rope, or of the height of the swing? I remember a very high childhood tree swing with light cable suspension which similarly wouldn't "swing"; Grandpa always had to push us even when we were elementary school-age. Can a swing be too high?
 A: It is probably the case that your swing is too long because the increase in the amplitude of a swing is achieved by changing the position of the centre of mass of the person on the swing relative to the position of the pivot.  
For a long swing the percentage change in the position of the centre of mass in relation to the length of the swing will be smaller and so the increase in amplitude for a given motion will also be smaller.  
There are a number of questions and answers about swings in this forum and a lot of article on the Internet.
There seem to be two principle mechanisms which increase the amplitude of the oscillations.  
One is to raise the centre of mass as the person on the swing passes the equilibrium position and then lower the centre of mass at an extreme.
Work is done by the person on the swing in raising the centre of mass and this manifests itself in an increase in kinetic energy.
This can be thought of as a consequence of the conservation of angular momentum $I_{\text{down}} \omega_{\text{down}} = I_{\text{up}} \omega_{\text{up}}$.
The literature seems to indicate that this mechanism is favoured when the swing amplitude is large.  
The other mechanism is to raise the centre of mass at an extreme and the work done by the person on the swing increase the potential energy of the system and hence the amplitude.
The literature seems to indicate that this method is the better one when the amplitude of the oscillations is small.
The movement of the centre of mass can be achieved in a number of ways.
Rotating the body so sometimes it is a vertical position and sometimes in a horizontal position; pulling on the suspension so that there is a kink in the suspension and then actually standing up and then squatting on the swing.
Perhaps it is also the heaviness and lack of flexibility of the chain which reduces the efficiency of lifting?
The production of a kink in the suspension method does not work if the suspension is rigid and so perhaps you could try and use strong but thin climbing rope instead?
A: You should push the swing correspond to its natural frequency. In the case of yours (swing), natural frequency of swing is $2\pi\sqrt{\large{\frac{g}{l}}}$ and for your swing, it is approximately 8 seconds. This means you should push the swing once per each 8 seconds. For starting, it is better that you release the swing from a height.
