Chosen Solution
My iMac is the model of iMac Intel 27" EMC 2429, mid 2011, which has 2 thunderbolt ports (yes, I cofirmed it is thunderbolt), which I have never tried to use until now. I tried the thunderbolt converting to VGA and it works fine. However, when I connect the imac to an external storage or converting it to USB 3.0, it does NOT work. The cable has no problem. It is a brand new cable bought from apple store, and it works fine on another mac book air. My question is, does the thunderbolt of this model of iMac really support external storage? Or it is my thunderbolt ports have some problem? I asked apple support, they said all thunderbolt ports should work for external storage inclusing this model.
I have two native Thunderbolt drives hanging off of my 27" iMac of the same series. So you either have a bad Thunderbolt adapter or there is something wrong with your logic board. I would start by getting a native Thunderbolt drive to try out if it works then the adapter is bad.
@danj is right, the model should work fine with external storage, especially if the same adapter and thunderbolt cable work on a different mac. It may not work, but resetting you PRAM is worth a shot to see if it’ll kick your thunderbolt port back into gear. Shut down the computer, hold down Shift+Command+P+R and press the power button. The computer should start up and chime, then shut off again. I always keep holding the four keys down until it goes through this process four time (just for good measure). Let us know if this works, or if you find another solution!
