The spindle with the chuck attached keeps falling out of my drill press. I can push the spindle (arbor shaft) up into the machine, but it later falls down. How do I fix this?

16 speed should have a Morse taper
for the std drill chuck

to fit a different chuck the morse taper needs to be emery sanded
and the replacement chuck should stick
after using the wood block pressure press method

if not you need a new morse taper for that spindle chuck


hint
i had my drill accuracy and off center spin tolerance down to one thou

with stretched old belts I got more speeds than 16
for special jobs
my 16 speed used a morse taper two for the drill chuck


this picture says its a spindle chuck
morse taper

More details about drill press go to the link Best Drill Press For DIY Professionals In 2018

this happens when the surfaces of the tapers both external and internal are damaged or have burrs
use fine emery paper to clean the surfaces
then put the chuck into the spindle taper and using a piece of wood pull the feed handle down rapidly so that the chuck is hit back up into the spindle

The cheuk is a morse taper (like a cone shape) fit to the shaft, if it comes loose just take a brass faced hammer and whack it back on.

The chuck is held in by the tapered shaft and friction. Make sure the tapered shaft on the chuck is clean with no burrs. Do the same for the socket side in the spindle. Swing the table out of the way and set a wood block under the chuck. Bring the spindle/chuck down into the block quite firmly which should lock the two tapered parts together. Hope this helps as it worked on my drill press!

Look at eReplacementParts.com and the parts diagrams

No there isnt. the chuck has a taper fit and needs to be cleaned and installed correctly. Use something like a brake clean or carb cleaner and spray the shaft and the internal socket of the chuck, wipe both clean with a paper towel and then move the table out of the way and with the chuck jaws fully retracted use a piece of wood and "seat the chuck" I usually strike the wood a couple of times. this should work as long as the tapers are not that worn.

If the spindle has a slot in it above the chuck and you can see the top of the chuck arbor in that slot then you need to get a wedged piece of steel commonly called a drift , insert in the slot so it bears on the top of the arbor and the top of the slot and hit it with a hammer. This will knock the tapered arbor loose from the spindle and it should fall out. Hope this helps you.

There is a long tapered hole in the spindle of the press known as a Morse taper. It holds a 6 or 8 inch morse taper arbor which is attached to the drill chuck. This is made to be removed frequently by putting a wedge shaped key into a horizontal hole in the drill press and giving it a little tap. This is for making tool change that involve removing the chuck. If you had two such tools (say two different style chucks) they would each have their own dedicated morse taper arbor. But there is also a short tapered cyclinder on the chuck end of the morse taper arbor, with a matching hole in the back of th chuck. This is not intended to come apart frequently, but it can be disassembled. I'm not sure which one is coming apart, but regardless of which one it is, you might try this fix. Clean up any bumps at all on the oustide of the taper and inside the hole with sandpaper 200 - 300 grit. Use a dowel to hold the sandpaper in the hole, and use the motor to spin it. --- JUST BE CAREFUL that if it grabs that it doesn't grab you! Once it's all cleaned up, you can you the press aspect of the drill press to seat it in place. You also might have to replace the arbor. They don't cost all that much and are available from Grizzly.com. You specify the particular taper for BOTH ends of the arbor. The long end is a Morse 2,3 or 4, and the short end is specific to the drill chuck manufacturer, like JT-2 (Jacobs chuck taper 2). Write back if you need more help.

While roughing up with sandpaper might work, but the usual problem that causes a chuck to fall off is a mismatched taper or either the chuck or the spindle has a dirty surface. The tiniest bloom of rust will cause this. Additionally a mismatch of depth will also cause a poor fit. To fit correctly the taper must be correct on both parts and they need to be clean and lightly oiled.

If it is necessary to san these pieces, use a very fine wet or dry paper. I would not use paper beyond 320 and use a light oil such as kerosene or diesel oil rather than water. Check the dept by looking inside at the wear marks. If the inside of the chuck receiver shows wear so close to the bottom that you cannot ascertain any distance between the wear marks and the bottom, the chuck may be bottoming instead of tightening on the taper correctly. You can easily check it with some fine shim material or perhaps three to five thickness of aluminum foil. Cut five thicknesses to fit in the bottom of the chuck (not all chucks have actual bottoms and go clear through) but not on the walls that are tapered. Reinstall, and remove. Is the aluminum foil damaged/show contact marks? If so, did the chuck go on all the way and come off more easily? If so, the problem is bottoming. Bottoming is quite rare but can happen when different than spec chucks are used.

You might also examine the surfaces of both. They should look pristine with no scratches and marks. If not, that could also be the problem. A correct taper fit is difficult to spin, or remove during normal drilling operations.

Good luck!