Timing needs adjustment is the most likely cause.
If there is a bad shaking after starting a 2005 Buick Lacrosse CXL, it could mean that a motor mount is broken in the engine compartment. The motor needs to be bolted down to the frame or it will vibrate out of its cradle in the engine compartment.
A bad valve or plug.
You may want to check the motor mounts, it sounds like the engine is vibrating.
go see a mechanic
If your engine backfires when trying to start and the valve timing is correct, then it could be electronic ignition unit or the pick up that causes the spark to occur with wrong timing.
Does it quit shaking when it warms up? What engine and year of Aerostar?
If the problem disappears when the shifter in placed in neutral it could be a transmission or engine mounts
There are countless different causes of an engine not starting, but in general, if the engine is receiving fuel, has ignition and compression, it will start. Begin with the basics. Next time it won't start, try a little starting fluid. If the engine runs briefly with starting fluid, you have a fuel delivery problem. Then go on from there. The only way to find a problem like that is to break it down. Check your Cam Sensors sometime they have a loose connection, your engine has 2 sensors a primary and secondary if the primary is the problem your engine wont start if its the secondary i will start but will run rough
there may be leakage in your coolant water cap at the time of starting vibration causes water to escape from it entering in to the engine causes white smoke
The Corolla could be shaking for many reasons. Start by checking the wheel bearings and brakes for wear. Then check the engine to see if it running rough, causing the shake.
The security is keeping it from starting which is what it is supposed to do.
Start by having wheel balance and tire wear thoroughly checked.