What can I do to automate shaking of a sauce bottle?

There are people who build their own shakers and from the looks of it, all you need is a motor (or a tool like a drill), a wheel and an arm with a joint that translate the rotation into oscillation.

Have a look at some Youtube videos of people who crafted a shaker from scratch:

• Simple wooden spray paint can mixer
• Model paint bottle shaker (I like this one best because it seems to have the most efficient shaking motion)
• 3D printed eliquid bottle shaker

Then there are other machines with oscillating motions that you can convert into a shaker:

• Handheld sewing machine shaker

• Fish tank air pump shaker
• Pepper grinder mixer (You can build a similar mixer with a drill on low speed)

• Kitchen aid mixer (german audio)

And of course there are ready-made products for this purpose, for example nail polish shakers, as reviewed in this video.

Which reminds me of those face cleansing brushes, that could also be adapted to hold and shake bottles. Make sure you get the oscillating kind, there are also ones that rotate or just vibrate.

Pack all the bottles securely in a box. Shake the box.

Forty 40ml bottles will only weigh a few pounds, and unless these bottles need serious shaking, an automated rig seems like overkill - surely you can spend 1 minute every few days shaking a box of bottles? If you do need to shake them longer and your arms are getting tired, consider other means of shaking - perhaps you could bounce the box on your knees while you watch TV, or toss the box in a backpack and go for a jog with it.

Your solution can be as simple as two muffin tins in an appropriate size.
As you fill each plastic sauce bottle, put it into the muffin tin.
When the muffin tin is full, put another tin upside down over the top to hold the bottles in place.

Pick up the whole thing and shake it a few times and all the containers' contents will be ready. You should be able to hold the assembly together with your hands using your fingers for the bottom tray and your thumbs to hold the top as you shake.

Use the muffin tin as a dispenser if you want. They are ready to be re-shaken by putting the top back on and re-shake as necessary, when necessary.

Muffin tins come in all sizes and shapes. You can use as many as you need for batches. You only need an extra empty muffin tin to use as a top for when you must shake or re-shake the bottles which remain clearly visible in the tray.

Good luck.

Chemical labs use devices made for this purpose. They're called shakers and they're available in many variations.

You can also get paint shakers, which will be cheaper but may need to be adapted to hold multiple small bottles instead of a paint can.

If you have to shake those bottle only once, do it manually.

If you have to do that every day, then it is more convenient to build an "agitating" rig. Attach a platform to 4 wheels, eccentrically (like a cart, but with wheels not centered). The eccentricity doe not need to be big - maybe 2-3 cm should be enough. Secure the bottles on the platform with another piece of board on the top.

Rotate one wheel - the entire setup will rotate up-dow, in a circular motion. To break the circular movement in the bottles, alternate the rotation direction.

To make it easier, attach a small motor. It will do the rotation for you, much faster.

The violet spots are the fixation points of the wheels and are in the center of the wheels. The red spots are the spots where the platform attaches to the wheels - and they are off-center.

You indicate you have a drill.. if you have a hand whisk too:

You can clamp it in the drill and use it to mix the sauce in a large container and then pour (or spoon) it into the bottles

Even a large number of 40ml bottles will weigh less than a load of wet washing, and I presume the lid is very secure (I guess it must be to give you confidence to attach them to a sawzall) you could consider putting batches of them all in a tumble drier on cool cycle. Maybe put some towels in there too, to soften the blows on the drum and to catch any spillages.

If you find they're still pounding the drum too much see if you can obtain a length of foam that is as wide as the drum is deep and as long as the circumference of the drum, then fit the foam around the circle of the drum before loading the bottles.. or put them inside several layers of pillow cases or other soft bag-like things and tie the bags closed

The more of them you load, the less they pound the drum (the ones at the top don't fall as far before landing on another bottle rather than whacking the drum) - maybe give a go empty first