Forces on the tire if it's just sitting there:
1)weight of the tire and the kid pulling down
2)support of the tree branch where it's tied on pulling the rope up
If the rope is in equilibrium, these forces will cancel.
Now if the swing is actually swinging, it's a bit more complicated. Suppose the tire is displaced by an angle theta and swinging with an angular speed omega.
Now the component of gravity along the line of the rope is:
Fg = m g cos(theta)
And you add to that the centripetal force required to keep the tire moving in a circle:
Fc = m omega^2 r
Add those up to get the tension in the rope, which will be the support required by the tree branch (neglecting drag and the weight of the rope itself):
Ftotal = m (omega^2 r + g cos(theta) ),
where m is the total weight of the kid and tire.
a child pushing his little sister on a swing
That would have to be perpetual motion
resonance
Gravity must be overcome.
Friction of air normally called drag (or air resistance), it could be argued that gravity also takes effect in stopping the swing, if it was only drag the swing could stop ten feet in the air, and if it was just gravity it wouldn't stop due to the laws of mass and momentum. The drag reduces the momentum, therefore making the swing slow to a stop and gravity keeps it as close to the ground as possible.
a child pushing his little sister on a swing
That would have to be perpetual motion
Probably the movement on a swing can be approximated by assuming that the magnitude of each swing will be a certain percentage of the previous swing (because of lost energy).
Because if the child falls off the swing, then they will just bounce on the soft rubber.
When the child is up in the air on either side of the climax of the swing, it builds potential energy to drop or swing back down towards the other side. When the child falls, gravity and the potential energy work together to make the child fall, and in turn, the kinetic energy not used for the fall goes into pushing the child back up on the other side. Say that there was no gravity involved in this situation. The child would only make it to the bottom of the swing, closest to the ground, because the amount of energy you store up in going higher cannot be amounted to greater as you swing down.
resonance
Start from the concept of what the structure is intended to do. What forces will act on the frame in its life? What materials are likely to be suitable. What is the strength needed for the service target? What size is allowed/desired. This will give you a rough idea of the type of approach to your question. Consider for example the design of a child's swing, and go to it!
Gravity must be overcome.
air resistance and friction in the bearings
Yes, swing sets have to pass tests before they are sold to consumers. If there is something defective, there will be a recall sent out on that swing immediately.
No
swing kids