C172 INSTRUCTOR PLT AND STUDENT HAD AN ENG MALFUNCTION THAT REQUIRED AN OFF ARPT FORCED LNDG.

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FLT INSTRUCTOR WAS TALKING STUDENT THROUGH A TRIM STALL WHILE STUDENT WAS PERFORMING MANEUVER. AFTER APCH CONFIGN WAS ESTABLISHED (FULL FLAPS; NO PWR; GLIDING); STUDENT APPLIED PWR TO CREATE TRIM STALL OCCURRENCE. AS PWR WAS APPLIED; THE C172 BEGAN TO SHAKE OR VIBRATE MODERATELY. FULL PWR WAS NOT ABLE TO BE ACHIEVED. PRIOR TO THIS HAPPENING; THERE WERE NO NOISES OR EXPLOSIONS TO INDICATE ANY TROUBLE. CARB ICE OR CONTAMINATED FUEL WAS ORIGINALLY SUSPECTED. C172 USES AUTO FUEL. THE C172 WOULD ONLY DEVELOP BTWN 1600 AND 1700 RPM MAX WITH MODERATE VIBRATION OF THE ENTIRE AIRFRAME. FOR THIS ACFT; MAX THROTTLE RESULTED IN A CONSTANT DSCNT OF BTWN 200 AND 400 FPM FLYING STRAIGHT. CORRECTIVE ACTIONS TAKEN WERE: CARB HEAT APPLIED FOR ICING IF PRESENT; EMER CHKLIST FOR ENG TROUBLE PERFORMED AND FUEL/AIR MIXTURE LEANED IN AN ATTEMPT TO GAIN MORE RPM'S. THE C172 WAS HEADED BACK TOWARD ZZZ ARPT VIA SAFE LNDG FIELDS (NOT DIRECT) WHILE THESE ATTEMPTS TO RESOLVE THE ENG PROB WERE BEING TRIED. AT ZZZ1; THE C172 HAD BEEN FLYING APPROX 8 MINS AND THE CONDITION HAD NOT IMPROVED; NOR HAD IT GOTTEN ANY WORSE WITH THE ENG AND VIBRATION. GOOD OIL PRESSURE WAS MAINTAINED THROUGHOUT THE ENTIRE TIME. WITH AN ALT OF AROUND 1800 FT MSL; POPULATED AREAS AND MOUNTAINOUS TERRAIN WAS ALL THAT LAY AHEAD WITH NO MORE SUITABLE LNDG FIELDS. THE FLT INSTRUCTOR DECIDED TO DECLARE AN EMER AND LAND THE ACFT IN THE FIELD AT ZZZ2; US. THE STUDENT SQUAWKED 7700 AND TALKED WITH ZZZ2 APCH CTL WHILE THE INSTRUCTOR CIRCLED THE FIELD AND MADE A SOFT/SHORT FIELD LNDG. A NORMAL SHUTDOWN OF THE SYS AND ENG WERE PERFORMED AFTER THE ACFT STOPPED. NO APPARENT DAMAGE WAS CAUSED TO THE ACFT BY THE LNDG AND NO PERSONAL INJURIES OCCURRED. ALSO; NO PROPERTY DAMAGE OCCURRED. THE ACFT WINGS WERE REMOVED AND THE ACFT HAULED BACK TO THE ARPT. THE FAA RESPONDED AND INVESTIGATED. THE ACFT IS TO BE INSPECTED BY MECHS/AI TO DETERMINE THE CAUSE OF THE ROUGH ENG AND LOSS OF PWR.