MLG FLC DELAYS CONTACTING CTR WHILE ABSORBED IN TROUBLESHOOTING AN ACFT VIBRATION.

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RECEIVED FREQ CHANGE TO ZDC. FO WAS FLYING AND I CALLED ZDC. JUST THEN THE ACFT BEGAN TO SHUDDER AND I SAID TO REDUCE AIRSPD BUT THE VIBRATION GOT WORSE. I SAID; LOWER STILL; BUT NO CHANGE. WE WERE NOW INDICATING ONLY 270 KIAS. AN MLG DOES HAVE A TENDENCY TO 'BUZZ' BUT ONLY AT SPD APCHING VMO/MMO; NOT AT THIS LOW SPD. WE HAD RELEASED OUT OF PHL WITH A 'PITOT HEAT INOP' LIGHT INOP (AS PER OUR MEL AND DISCUSSION WITH TUL MAINT BASE OF MY AIRLINE). MY PRIORITY CONCERN NOW -- WAS THERE A HIDDEN AND MORE SEVERE PROB IN OUR PITOT STATIC SYS THAN DIAGNOSED AND WERE WE GETTING ERRONEOUS SPD INDICATIONS CAUSING THIS UNUSUAL 'BUZZING.' AFTER TROUBLESHOOTING FOR A FEW MINS; I SATISFIED MYSELF AND CREW THAT THE PITOT WAS WORKING AND THE ACFT WAS FIT TO CONTINUE TO DFW. I THEN REALIZED ATC HAD NOT RESPONDED TO MY 'CHK IN' AND TRIED AGAIN. I WENT BACK TO PREVIOUS ATC FREQ; GOT CHEWED OUT; THEN SENT BACK TO ZDC. THIS ENTIRE EVENT WAS WITNESSED BY AN ATC CTLR BASED IN CRP. HER WORDS WERE SOMETHING LIKE; 'I HAVE NEVER SEEN A FLC SO BUSY.' IN CONCLUSION; I FEEL THIS EVENT WAS JUST TIMING. A FEW SECONDS EITHER WAY AND ATC WOULD HAVE BEEN TOLD OF OUR PROB OR TO STAND BY. OUR CONCERN WAS CORRECTLY FOCUSED ON OUR ACFT'S INTEGRITY. THIS EXACT SCENARIO IS WHAT CAUSED THE CRASH OF AN AIRLINE'S LGT.