What was found
PILOT REPORTED AFTER TAKE OFF, LT PROP RPM ROSE TO 1730 FOR 15 SECS, PULLED BACK PROP LEVERS AND CONTINUED TO CLIMB, LEVELED OFF AND TRIED TO SYNC PROPS, RECEIVED A VIBRATION, TURNED OFF PROP SYNC AND SYNC'D, NR 1 AT 1400 AND NR 2 AT 1550, STARTED LOSING PROP RPM, WAS INSTRUCTED TO SHUTDOWN ENGINE, EMERGENCY DECLARED. MAINTENANCE REPLACED THE NR 1 PROPELLER RPM INDICATOR IAW BMM 31-10-00.