FLC OF ACR MLG ACFT EXPERIENCED A MALFUNCTION IN THE FMS VNAV ACFT SYS AND OVERSHOT CLB ALT.

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ON CLBOUT FROM ORLANDO; WE WERE FOLLOWING A NORMAL VNAV PROFILE; 330 KTS; CLB RATE 2500 FPM. AT ABOUT 20000 FT; CTR ASKED FOR 300 KTS. THIS INCREASED OUR CLB RATE. PASSING 21300; THEY REQUESTED A LEVEL OFF AT FL220 IT WAS SET IN AND THE FMA ANNUNCIATED ALT CAPTURE; BUT THE AUTOPLT MADE NO IMMEDIATE ATTEMPT TO LEVEL OFF. A 300 FT OVERSHOOT WAS DEVELOPING WHEN CTR CLRED US TO FL230. THE ACFT CLB RATE CONTINUED TO INCREASE. IT BECAME APPARENT THAT WE HAD EITHER A TRIM RUNAWAY OR AN FMS VNAV PROBLEM. THE AUTOPLT AND AUTOTHROTTLE WERE DISENGAGED. NOSE UP TRIM WAS HIGH; AND STICK PRESSURE REQUIRED TO INPUT NOSE DOWN WERE SIGNIFICANT. THROTTLES WERE PUT AT IDLE. A MILD TURNING WING DOWN MANEUVER (IE; WING OVER) WAS COMPLETED TO BRING THE ACFT UNDER CTL WITHOUT INJURY TO CREW OR PAXS. THE CAPT DID NOT WISH TO DO A SUDDEN OR 0 G MANEUVER. THE ENTIRE TIME THE CAPT WAS RUNNING NOSE DOWN TRIM. CTR WAS INFORMED OF OUR PROBLEM. WE LEVELED AT FL240 OR 1000 FT HIGH. THERE WAS NO CONFLICTING TFC AND CTR SAID NOTHING OF OUR ALTDEV. THEY GAVE US A CONTINUED CLB TO OUR CRUISE ALT.