jestern77 I tried powering the Beagle Bone without SD card and BELA but it doesn't work. Just a little blue light in the power led. But then it doesn't start. Not a good sign. Is that recoverable?
Doesn't look promising. I'd suppose it's dead. You could try to remove the Bela cape and see if it boots without it. If you have a serial cable, you can try to plug it into the male header on the BeagleBone and see if you get any info there (see here for different cables/pinouts). Also, try flashing a new SD card and boot with it, powering the board while pressing the "USR" button on the BeagleBone. If it doesn't boot in any of these cases, and it doesn't show any info on the serial port, then it's dead for good.
jestern77 Can you help me check what is salvageable? Thanks. Tell me maybe which is the component I burned so I can try to replace it.
So that's FB1, it's a 0603 ferrite bead used to decouple the noise between the digital and analog ground. In principle, you could replace it with a short circuit, and get worse noise performance but still working fine. You can also order a replacement from, e.g.: Farnell.
What concerns me the most is: how did you manage to burn that? Will anything else have been damaged in the process? If the BeagleBone died, then it's reasonable to think the cape will have died as well. If the BeagleBone turns out to be dead, and you replace it, then you can try the following to see if the Bela cape is still recoverable: remove the cape and try to measure the resistance between P9_01 and P9_03, or P9_01 and P9_05, or P9_01 and P9_07 of the CAPE. If none of those is a short circuit, then measure the resistance between P9_07 and every other pin on the cape: only P9_08 should be a short circuit, all the others should have some rather high (possibly infinite) resistance. If this test passes, then it's safe to plug the cape on the new BeagleBone, after replacing the ferrite bead, to evaluate whether it still works.
Here are the schematics and board diagram (you need the free edition of EAGLE to view them, but there are also pdf renderings), and the bill of materials is also there: https://github.com/BelaPlatform/bela-hardware/tree/master/cape .