DA20 KATANA PLT EXPERIENCES UNUSUAL NOISE AND VIBRATION ON LNDG AND DISCOVERS THAT THE NOSEWHEEL HAD COME OFF ON TKOF.

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THE NOSE GEAR SEPARATED FROM THE ACFT UPON TKOF. I (PLT) WAS UNAWARE OF THAT. UPON LNDG ZZZ THE PROP AND REMAINING NOSE GEAR ASSEMBLY WAS DAMAGED. THIS EVENT OCCURRED AFTER THE SECOND LEG OF A SHORT DURATION FLT. THE PURPOSE OF THE FLT WAS TO PURCHASE FUEL FROM THE SELF-SERVE FACILITY AT A NEARBY ARPT ZZZ1. PREFLT INSPECTION WAS CONDUCTED. NOTHING UNUSUAL WAS NOTED. AFTER FUELING AT ZZZ1; THE RTN LEG TO ZZZ WAS THE SAME. SPECIFICALLY; TAXI-OUT; RUN-UP; TKOF; FLT; AND LNDG WERE UNEVENTFUL. THE LNDG WAS RELATIVELY SOFT AND ON THE MAINS. THE SUBSEQUENT ROLL AND TAXI WERE UNEVENTFUL. I DO NOT RECALL ANY BUMPS OR STEERING DIFFICULTY. TAXI-OUT; RUN-UP; AND TKOF ROLL WERE UNEVENTFUL. I DO NOT RECALL ANY BUMPS OR STEERING DIFFICULTY. I SPECIFICALLY DO RECALL BEING PLEASED WITH MYSELF FOR HOLDING THE CTRLINE OF THE RWY DURING THE ROLL. DURING FLT THERE WERE NO UNUSUAL SOUNDS; VIBRATIONS; FLT CHARACTERISTICS; OR THE LIKE. THERE WERE NO UNUSUAL SOUNDS; VIBRATIONS; FLT CHARACTERISTICS; OR THE LIKE DURING THE APCH AND PATTERN. AFTER TOUCHDOWN ON THE MAIN GEAR; UPON LOWERING THE NOSE THERE WAS UNUSUAL SOUND AND VIBRATION. UPON STOPPING THE ACFT; SHUTTING DOWN THE ENG; AND EXITING THE ACFT; IT WAS IMMEDIATELY APPARENT THE NOSE GEAR WAS MISSING. THE BOLT THAT HOLDS THE NOSE GEAR WAS SUBSEQUENTLY DISCOVERED AT THE DEP FIELD ZZZZ1. IT WAS SHEARED ABOVE ITS CASTLE NUT WITH COTTER PIN INTACT. ANOTHER PLT FOUND THE NOSE GEAR ADJACENT TO RWY 31 AT THE DEP ARPT. THE ACFT WAS WITHIN BOTH ITS ANNUAL AND 100 HR INSPECTIONS.