DURING A LANDING AT A LOCAL AIRPORT, THE PILOT REPORTED THAT HE EXPERIENCED VIBRATION IN THE AIRPLANE ON A PREVIOUS LANDING, WHICH HE FELT ORIGINATED IN THE TAILWHEEL. THE PILOT AND A MECHANIC TIGHTENED THE TAILWHEEL NUT. THE PILOT ELECTED TO PROCEED TO HIS NEXT DESTINATION AND DURING THE LANDING THE VIBRATION RECURRED. DURING THE LANDING ROLL, THE AIRPLANE VEERED OFF THE RUNWAY AND GROUND LOOPED. AN EXAMINATION OF THE TAILWHEEL REVEALED NO ANOMALIES.