2000 Grand Am engine shakes at start up

Automatic transmission engine is 3.4
The engine shakes at start up and there is a noise coming from left side of engine where the pulleys and the serpentine belt are that sounds almost like something is "knocking" together (noise is not coming from the engine itself as far as I can tell). This only happens at start up, rpms are not affected and it does not happen anymore after I get the car going. I have driven up the road about 5 blocks and pulled off side of road put it in park and it is fine. It also seems that when car is in reverse it is worse and neutral it is almost not there at all. Sometimes after driving it and shutting it down for a few minutes it is fine when I restart it. Other times it acts up. It always acts up after it has sat for more than 30 minutes. I just replaced the LIM gasket and water pump, torqued everything back down to specs. Dont know if that has any bearing on the situation or not. Noise and engine shaking started happening about 3 or 4 days after I got the LIM job complete.
TY in advance

Start engine, shift into reverse, pull E-brake open the hood and try to see what is making that noise. My best guess would be it is engine mount related, (mount not centered or it is loose) did you have any extra bolts left after working on the engine? When you shift through gears you will notice that engine is moving sometimes towards and sometimes away from the radiator. Depends if you shift into D or R. So it is possible that when you shift into R whatever is loose is getting closer to hit whatever it is hitting and making that knocking sound.

motor mount is centered and torqued i will try your suggestion and see if i can get better idea of exactly where sound is coming from
I had no bolts or anything left over
oh and i forgot to mention that if i just let the car run for awhile both the sound and the motor shaking go away. It just takes longer sometimes than it does at other times
ty

well i didn't notice much movement toward or away from the radiator. Seems to be just like it always looked in the past. And the noise seems louder closer to the firewall.

With engine off try to shake the top of the engine by hand and listen for the knock. It can be exhaust pipe or exhaust shield hitting firewall.

well have shaken the engine by hand and no noise :( was hoping i would find out what was wrong by doing that. Other than the motor mount any other ideas as to what could cause the engine to shake at start up?

can you upload the video of the problem and post your link here it would be easier that way.

- also when you say at start up is it only while you are cranking or once engine is started the problem exists for several seconds?

problem is only after engine has started I am leaning more on the side that it is not getting gas/air ratio correct at start up. everything else it could be seems as if it would persist whenever I got back down to idle speed. But car is fine after the initial shaking at start up. I will try and get video posted, only have camera that does video so here is hoping to the quality not being garbage.

i would change the spark plugs and wires it might help