Well, there's a pretty solid reason why an actual artillery piece (as in a real cannon and not an overgrown arquebus or swivel-gun) would never work on the back of an elephant, or any other beast of burden, even if you could train the animal to not go bonkers at the infernal noise.
Recoil.
Seriously. Those things had *no* compensation save for the weight of the gun and its frame. None. El zippo nada. Obviously you're going to hit the weight limit of how large a piece even an elephant can carry in the first place, nevermind now the ammo and gunpowder for the damn thing, and even large heavy guns on massive frames kicked like a dozen cast-iron mules and tended to move noticeably from their original position as a result. You fire a real cannon atop an elephant, and a fairly immediate result will be the heffalump suddenly being relieved of a considerable burden when the gun, its carriage and most likely large parts of the fighting-platform or -tower as well as several hapless cannoneers crash on the ground near its tail...
Muskets on elephant-back worked, though. The Timurids were eventually chased out of Central Asia by the Uzbeks and carved themselves an empire in northern India under the name Moghuls (a name pretty obviously derived from "Mongol" - recall that old Timur had claimed, AFAIK not wholly unjustly, dynastic descent from Genghis himself), and at some point some clever fellow put an armoured arquebusieur onto the fighting-platform apparently with fairly satisficatory results. I wouldn't put it past someone to have tried light swivel-guns or very large arquebuses akin to the big nonportable lugs Europeans sometimes employed in siege defenses up there too, but that may not have worked as well. I can't claim to know too well how those war elephant howdahs were built, but the railings probably weren't quite the sturdiest recoil-absorbers around.
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