C182R pilot reported hearing a loud noise accompanied with engine vibrations during climb. After landing it was discovered the propeller dome assembly was missing.

Narrative

Departing ZZZ to the NE 5400 climbing 5500. A loud clunk was heard; aircraft physically shuddered once; and vibrations synchronous with engine RPM immediately occurred. The control yoke was still responding; PIC reduced manifold pressure to partial power which reduced vibrations. A burning smell was then detected in the cockpit. Immediately turned direct ZZZ and started descending as the ForeFlight glide rings showed we could easily glide back to ZZZ with thousands of feet of altitude to spare. Engine instruments were all normal. Made CTAF call to clear runway as there were other aircraft departing behind us. Unicom on the ground reported they were helping clear the runway. Tested throttle; power could be obtained but vibration still present. Arrived over ZZZ at 3200' and circled down in one tight left pattern; descending. Unicom reported runway clear. 2;500 lowered gear on downwind; flaps 20; turned final with lots of altitude so flaps 40 degrees to descend steeply to runway surface 20. Unicom reported Dome Assy-Spinner missing from aircraft while taxiing. Ramped aircraft; chocks; tied down; and reported to aircraft maintenance. Dome Assy-Spinner gone. Half of each Bulkhead Half Assy torn or cracked off; 1 of 2 Connector-Bulkhead Half gone. Half of each Bulkhead Half Assy gone likely causing unbalance and vibration. Half of each Bulkhead Half Assy that was left was bent and rubbed on the cowl; likely causing heat and the burning smell. The half of each Bulkhead Half Assy still in place shows that all Dome Assy-Spinner mounting locations were ripped out of the half of each Bulkhead Half Assy; indicating those Bulkhead Assy-Spinner Attaching Screws and Washers were in place. No fragments of the actual Dome Assy-Spinner were still present. Suspect failure was mostly the two Bulkhead Half Assys. Suspect Bulkhead Half Assys cracked near the Attach Plate and the uncracked portion of the Bulkhead Half Assys could not hold the Dome Assy-Spinner in place and it tore the rest of the way loose from the Bulkhead Half Assys at the Bulkhead Assy-Spinner Attaching Screws location.A thorough preflight plus additional checks and a runup according to the POH was performed by the PIC before departure. During pre-flight the external surface of the Dome Assy-Spinner had no visible cracks or dents; each and every Bulkhead Assy-Spinner Attaching Screws and Washers that hold the Dome Assy-Spinner on were in place and not loose; all rivets holding the front Support - Spinner plate in place were not loose; the entire Dome Assy-Spinner was tight and not loose on the end of the crankshaft; and the prop was not touching the Dome Assy-Spinner. PIC has no known way to visually inspect the Bulkhead Half Assy for cracks near the Attach Plate on pre-flight check because they are physically covered up by the Dome Assy-Spinner.