Some hatch mechanics 📐🔨

I am often a DIY’er and I have a really heavy hatch (32 kg) at home that I want to open automatically with an electric actuator. Something like the hatch in the image below.

You can buy these actuators for 50 to 100 euros online: but which one do I need? or:

• What is the minimal and maximal length (and stroke) that I need?
• What is the force the actuator should be able to push?

I made a schematic model, with

• $l$ is the length of the hatch ($l=0.75 m$ in my situation)
• $a$ and $b$ are yet unknown attached positions of the actuator from the hinch
• $\varphi$ is the the angle of the hatch between 0 and 90 degrees
• $\theta$ is the angle of the actuator with the vertical $\theta$
• $w$ is the dynamic actuator length, dependant on the hatch position.

Actuator length

With some geometry and pytagorasch we can calculate $w$ dependant on hatch position:

$$w^2 = \left(a\cdot\sin\varphi+b\right)^2+\left(a\cdot\cos\varphi\right)^2$$

From which follows that $w$ varies between the retracted length:

$$(\varphi = 0) \rightarrow w_\text{retracted} = \sqrt{a^2+b^2}$$

and the extended length:

$$(\varphi = 90) \rightarrow w_\text{extended} = a+b$$

The minimal stroke length is the difference:

$$\Delta w = a+b-\sqrt{a^2+b^2}$$

Force on the actuator

The weight of the complete hatch is $W = m \cdot g = (32) \cdot (10) = 320 N$. From the moment around the hinge, you can calculate the normal force on the actuator:

$$F_x = \frac{Wl}{2a \cdot \cos \theta}$$

Which shows that the force in the actuator is the largest where $\theta$ is the smallest, which is when the hatch is closed ($\theta=0$). With some more geometry we find the relation between $\theta$ and $\varphi$ as:

$$\tan \theta = \frac{a \cdot \cos \varphi}{a \cdot \sin \varphi+b}$$

So the maximum force is where:

$$\tan \theta = \frac{a}{b}$$

Then simplify, with $\cos \left(\arctan (x)\right) = \frac{1}{\sqrt{x^2+1}}$; we can rewrite the solution for the maximum force to:

$$F_{\text{max}} = \frac{Wl}{2} \sqrt{\frac{1}{a^2}+\frac{1}{b^2}}$$

Choose a product

Luckely, there are many suppliers. As a ballpark figure, when $a=0.35m$ and $b=0.25m$:

• $w_\text{retracted} = 0.43m$
• $w_\text{extended} = 0.60m$
• $F_\text{max} = 590 N$

Here a list of products that meet my requirements