CHEYENNE II XL ON NIGHT TRAINING FLT. WAS MAKING GAR ON 1 ENG FROM ILS RWY 5 APCH. IN ATTEMPTING TO GET THE ACFT CLBING; FAILED TO MONITOR HDG AND DEVIATED 40 DEGS TO THE N. TWR QUESTIONED HDG AND AT THAT TIME ACFT WAS BEGINNING TO CLB. INSTRUCTOR GAVE THE ENG BACK AT 1000 FT.

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I WAS WEARING FOGGLES WHILE PRACTICING THE ILS RWY 5 APCH TO LAKELAND ARPT IN VMC CONDITIONS AT NIGHT. AS I WAS DSNDING TO 1700 FT MSL AND INTERCEPTING THE LOC COURSE; THE FLT INSTRUCTOR ON BOARD REDUCED THE L ENG'S PWR TO IDLE TO SIMULATE A FAILURE. I HAD ALREADY INTERCEPTED THE GS AND HAD THE GEAR AND APCH FLAPS EXTENDED. AFTER INCREASING PWR TO THE R ENG; I FOUND THAT I WAS ABLE TO MAINTAIN AIRSPD AT 120 KTS; AND REMAIN ON THE GS. THE ASYMMETRICAL THRUST CAUSED ME TO MOVE TO THE L SIDE OF THE LOC COURSE; BUT I WAS ABLE TO RE-CTR THE NEEDLE BEFORE REACHING THE OM. BY THIS TIME I HAD IDENTED THE DEAD ENG AND ADVISED THE INSTRUCTOR THAT I WOULD NORMALLY FEATHER IT. HE SAID TO LEAVE THE PWR WHERE IT WAS (AT IDLE) AND PLAN TO EXECUTE A SINGLE ENG GAR AT DECISION HT. HE SAID I COULD USE THE L ENG IF IT BECAME NECESSARY. AT THIS POINT I ASSUMED THAT HE WAS EXPERIENCED ENOUGH TO REALIZE THAT WITH JUST THE 2 OF US AND PARTIAL FUEL; THIS ACFT (A CHEYENNE II XL) COULD CLB WITH 1 ENG WINDMILLING. AT DECISION HT OR ABOUT 400 FT MSL; I RAISED THE R ENG TO FULL PWR AND INCREASED R RUDDER PRESSURE WHILE PITCHING UP SLIGHTLY AND PRESSING THE GAR BUTTON ON THE THROTTLE. I CHOSE TO IGNORE THE PITCH COMMAND ON THE FLT DIRECTOR; BUT TRIED TO KEEP THE DELTA PARALLEL TO THE COMMAND BARS. THE ACFT REMAINED AT 400 FT INDICATED ALT SO I RAISED THE FLAPS AND PITCHED UP A LITTLE MORE. I WAS STILL HOLDING ALT AND AIRSPD WAS BLUE LINE SO I RAISED THE GEAR. AT SOME POINT DURING THE GAR THE INSTRUCTOR BEGAN CHANTING; 'GET IT CLBING;' AS I PROBABLY WOULD HAVE DONE HAD I BEEN IN HIS PLACE. HAVING RAISED THE GEAR I FELT CONFIDENT THAT THE ACFT WOULD BEGIN CLBING ANY SECOND SINCE I WAS STILL AT BLUE LINE AIRSPD. I WAS ALSO SURE THAT I WAS TRACKING RELATIVELY STRAIGHT DOWN RWY 5 SINCE I WAS STILL KEEPING THE DELTA PARALLEL TO THE COMMAND BARS. AT THIS POINT THE TWR OPERATOR; WHO HAD ASKED US TO FLY RWY HDG DURING THE MISSED APCH; CAME ON THE RADIO AND ASKED IF WE HAD TURNED N. MY INSTRUCTOR MADE A REPLY WHICH I DID NOT HEAR BECAUSE MY ATTN WAS NOW ON OUR PICTORIAL NAV INDICATOR WHICH I REALIZED HAD BEEN OMITTED FROM MY SCAN SINCE THE BEGINNING OF THE GAR. AT THE SAME TIME I NOTICED WE HAD BEGUN TO CLB SO I INITIATED A GENTLE TURN TO THE R TO GET BACK ON COURSE. I THINK I HAD YAWED APPROX 40 DEGS TO THE L SINCE ADDING PWR FOR THE GAR. UPON REACHING 1000 FT MSL; THE INSTRUCTOR TOLD ME TO GO AHEAD AND ADD CLB PWR TO THE L ENG. WE CLBED TO A HIGHER ALT TO LEAVE LAKELAND'S AIRSPACE AND CONTINUE THE LESSON. IT DIDN'T OCCUR TO ME UNTIL I WAS ON MY WAY HOME FROM THE LESSON AND REHASHING EVENTS IN MY MIND THAT WE MAY HAVE PASSED VERY LOW OVER THE TERMINAL OR SOME OF THE HANGARS AT LAKELAND WHILE I WAS TRYING TO MAKE THE ACFT CLB. THINKING BACK; THE PROB WITH HDG CTL WAS NOT DISCOVERED UNTIL THE TWR OPERATOR ASKED ABOUT OUR HDG. I SUSPECT THE INSTRUCTOR'S ATTN WAS AS FOCUSED ON OUR ALT AS MINE. I DON'T BELIEVE HE NOTICED THE CHANGE IN HDG EITHER INSIDE OR OUTSIDE OF THE COCKPIT. THE DARKNESS OF THE NIGHT COULD HAVE CONTRIBUTED TO HIS LACK OF AWARENESS. IT WAS A MISPERCEPTION ON MY PART WHICH CONTRIBUTED TO THE TURN. I ASSUMED THAT KEEPING THE DELTA PARALLEL TO THE COMMAND BARS WHEN IN GAR MODE WOULD KEEP US ON RWY HDG; SO I OMITTED THE PICTORIAL NAV INDICATOR FROM MY SCAN DURING THE SECONDS LEADING UP TO THE CALL FROM THE TWR. ALTHOUGH MY HDG BUG WAS SET ON 50 DEGS; I HAD NOT PRESSED THE HDG HOLD BUTTON ON THE MODE CTLR. ANOTHER ASSUMPTION I MADE WAS THAT THE FLT INSTRUCTOR WAS LOOKING OUTSIDE AND WOULD ADVISE ME TO ADD PWR TO THE L ENG BEFORE WE BECAME TOO CLOSE TO ANY OBSTACLES ON THE FIELD OR AT THE END OF THE RWY. I DIDN'T ANTICIPATE HIM BECOMING DISTR (IF HE WAS). WE HAD BOTH EATEN RECENTLY AND WERE PHYSICALLY RESTED SINCE WE HAD SPENT TIME ON THE GND DISCUSSING THE ACFT MANUAL AND SOME FARS PRIOR TO THE FLT. WE HAD NOT DISCUSSED SINGLE ENG GARS PRIOR TO THE FLT; BUT I HAD RECENTLY REVIEWED THE PROC IN THE PLT'S OPERATING HANDBOOK. IN RETROSPECT; I DON'T BELIEVE IFI WERE THE INSTRUCTOR DURING THIS LESSON THAT I WOULD HAVE ASKED MY STUDENT TO PERFORM THIS PROC CLOSE TO THE GND (AT LEAST NOT THE FIRST TIME AROUND AND WITHOUT HAVING DISCUSSED IT PRIOR TO THE FLT). I WISH I HAD DECLINED TO INITIATE A SINGLE ENG GAR WITH A PROP WINDMILLING; BUT I LET MYSELF BECOME TOO FOCUSED ON THE ASSIGNED TASK TO CONSIDER IF IT WAS A GOOD IDEA. I WAS SKEPTICAL; BUT NOT VOCALLY. IN THE FUTURE; I WILL NEITHER TRY TO PERFORM THIS PROC CLOSE TO THE GND NOR ASK ANY OF MY OWN STUDENTS TO DO SO. THIS SHOULD ONLY BE ATTEMPTED IN A FLT SIMULATOR; AT ALT; OR DURING A REAL EMER WHEN IT IS THE LAST AND ONLY ALTERNATIVE.