What was found
(CAN) ON DEPARTURE, AND SLIGHTLY AFTER V1, THE FLIGHT CREW FELT A MILD VIBRATION. THE NOSE TIRES WERE OFF THE RUNWAY AT THIS POINT, GIVEN WE WERE IN THE MIDDLE OF ROTATION. BASED ON THE VIBRATION AMOUNT, ONE OF THE SHIMMY DAMPERS WAS SUSPECTED. THEY CONTINUE WITH A NORMAL DEPARTURE PROFILE, ENGINE INDICATIONS WERE NORMAL. THE CREW DID NOT HEAR ANY LOUD NOISES. ROUGHLY 30 MIN LATER, ATC CALLED TO RELAY A MESSAGE THAT ANOTHER ACFT NOTICED RUBBER ON THE RUNWAY. THEY RAN THROUGH THE APPROPRIATE CHECKLIST. AFTER AN ASSESSMENT OF THE SITUATION, ON THE GROUND AND IN THE COCKPIT, THEY HAVE DECIDED TO CONTINUE TO DESTINATION. AT THIS POINT, THEY BRIEFED THE F/A ON THE SITUATION AND THE FACT THAT THE CFR WOULD BE REQUESTED FOR LANDING. THAT IS WHEN THEY FOUND OUT THAT THEY HAD HEARD A LOUD NOISE AND A VIBRATION. ABOUT 40 MIN. OUT FROM DESTINATION, EMERGENCY WAS DECLARED TO MAKE SURE CFR WOULD BE STANDING BY. UPON LANDING THERE WAS VIBRATION ALL THE WAY TO TAXI SPEED. WHILE IN THE ROLLOUT THE F/A CALLED UP AND NOTIFIED ME THAT SPARKS WERE COMING OFF THE CAPTAINS SIDE OF THE ACFT. CFR INSPECTED THE AIRCRAFT AND FOUND 1 BLOWN TIRE, CAPTAIN SIDE INBD. DISCOVERED DURING THE POSTFLIGHT INSPECTION WAS DAMAGE TO THE FLAPS AND CARGO DOOR.