Mike,
In my opinion, most shutdown problems are due to a fault in the hardware (nine times out of ten it's either video, RAM or power supply issues)....
That said, the reason they tell you to try this and this and this and then this, and after that this etc... is because it's difficult to pinpoint.
You have to start at step one, rule it out and then move on until you hit the actual cause. If there were an error you might have an easier time distinguishing the problem... but random reboots are hard to figure out.
First place I would start would be
to visually inspect all connects, fans, jumpers, etc... make sure that the case has good airflow, that the CPU, Power Supply and/or video card fans are working properly and free of any obstructions.
I'd reseat the memory just to make sure that a stick hasn't wiggled loose enough to give your machine a reboot syndrome.
Second place I'd go with memory tester:
http://www.memtest86.com/
After you've checked and confirmed that the memory is either good or bad, (if bad you've discovered the problem)... if good, I'd move on to swapping the power supply.
If changing the power supply doesn't do it, I'd then test the video card (if it's external)... Try exchanging it with a known working unit, or trying it out with another computer.
I'd then look at updating the OS (try that SP2 for windows 98 that I posted)... that could fix it.
Another thing that might help is a BIOS update. But I'd be wary of this due to the fact that you can literally brick the computer making it not function... and becoming a huge paperweight.
It could be a serious problem like a motherboard malfunction. In that case it might be necessary to change the motherboard completely... AFAIK there's no real way to determine if the motherboard is the culprit without trying to exhaust all the other possible problems....
Bookmarks