Angles on swing sets

Question

I'm building a swing set for my children. All of the designs I've seen involve building two A-frames and connecting them at the top with a crossbar/beam from which hang the swings. The A-frames are placed perpendicular to the ground. In addition, it seems that the A-frames are commonly constructed at an angle of 12 degrees from the perpendicular to the ground.

I have several questions:

1. Where does the 12 degree angle come from?

2. I've been on swingsets that move too much. I think this is due to a combination of unanchored posts and the angle between the legs of the frame being too little. How do I compute the angle that will provide optimum stability?

3. Shouldn't the frames be placed at an angle with respect to the ground (e.g. 70 degrees) rather than perpendicular, in order to prevent side-to-side motion?

4. How would I go about computing the above?

Assumptions that I'm going on include:

• 2 swings
• Adults will swing (up to 200 lbs)
• A-frames will be anchored to concrete piers

Answer 1

Using a simple pendulum approximation, at the point of , the maximum tipping force occurs when the swing is at $\theta=45°$ angle. In general the forces on the bar are

$$ A_x = -\cos\theta \left( m g \sin \theta + \frac{\tau}{L} \right)-m L \dot{\theta}^2 \sin \theta $$ $$ A_y = m \cos \theta \left( g \cos \theta + L \dot{\theta}^2 \right) - \frac{\tau}{L} \sin \theta $$

where $L$ is the chain length, $\tau$ is the torque applied by your back on the pelvis (pull up torque) and $m$ is the mass of the swinger.

If $h$ is the height of the bar from the ground, and $b$ the base distance across the A-frame then the forces on supports are

$$ G_x = -A_x $$ $$ G_y = \frac{1}{2} A_y - \frac{h}{b} A_x $$ $$ H_y = \frac{h}{b} A_x + \frac{1}{2} A_y $$

Combining you will see that the forces in the frame are not always compressive, but tensile also. In the above expression $H_y$ is always negative (frame pulling on the ground) when $\theta > (\text{frame angle})$.

So as far as physics goes, there is no significance to the 12°. Engineering wise, I like the other answer of minimizing the material used.

Answer 2

The greater the angle in the A-frame, the more metal you'll have to use. This, balanced with wanting to keep the swing set firmly planted probably led to the heuristic 12 degrees.

(EDIT: ignore gravity transmitted through the chain because it doesn't ever apply torque to swing set.) If you want to calculate some forces on the swing, the force a swinger exerts on the top bar is along the swing chains and equal to mass*speed^2/length of chain. The speed is minimal when the swing chains are parallel to the ground, and this is also when the direction of the swinger's force is delivering the maximum torque to tip over the swing set. From this argument, I think the tipping force is not much.