STUDENT PLT IN SMA SEL ACFT MADE HARD LNDG DURING A PRACTICE LNDG RESULTING IN DAMAGE TO THE NOSE STRUT AND A BLOWN NOSE GEAR TIRE.

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I WAS TEACHING A PRIVATE PLT STUDENT ON OUR THIRD FLT. OUR FLT OF MIN CTLABLE AIRSPD; STALLS AND STEEP TURNS WAS FINISHED AND WE WERE HEADED BACK TO THE ARPT (MID CONTINENT) FOR MULTIPLE LNDGS. OUR TRAINING HAS BEEN ACCELERATED BECAUSE THE STUDENT HAS SEVERAL FRIENDS WITH ACFT HAS FLOWN; BUT HAS NOT HAD ANY FORMAL TRAINING. THE SMA ACFT HAS BEEN OUR ACFT OF CHOICE. THE ACFT IS IN GOOD CONDITION BUT THE NOSE STRUT HAS GONE FLAT RECENTLY AND WAS REPAIRED (FILLED). THE NOSE TIRE WAS WX WORN. ON OUR SECOND LNDG; THE STUDENT'S LNDG FLARE WAS HIGH AND I PROMPTED HIM TO ADD PWR WHICH HE DID. HE DID NOT CONTINUE TO INCREASE ANGLE OF ATTACK TO REDUCE HIS VERT DSCNT. WE TOUCHED DOWN AS I WAS INCREASING ANGLE OF ATTACK AND PWR. IT WAS A 3 POINT LNDG; VERY ABRUPTLY WE BOUNCED UP AND I RECOVERED; TOUCHING DOWN ON THE MAIN GEAR. WE ROLLED OUT AND THE NOSE TOUCHED DOWN. THE VIBRATIONS WERE FROM THE FLAT TIRE. NOSE STRUT WAS ALSO FLAT. WE NOTIFIED TWR AND TAXIED CLR BUT NOT TO THE HOLD SHORT LINE. HE ASKED IF WE COULD TAXI CLR OF HOLD SHORT LINE. WE DECLINED BECAUSE I DIDN'T WANT TO DESTROY THE RIM BY TAXIING ON IT. WE SHUT THE ACFT DOWN; PULLED THE ACFT CLR; NOTIFIED THE TWR WE WERE CLR. THEY CALLED FBO. WE TOWED THE ACFT TO FBO. THE TIRE WAS REPLACED AND THE STRUT REFILLED. THERE WAS NO OTHER DAMAGE. THE TWR DID HAVE TO CLOSE THE RWY FOR A FEW MINS UNTIL WE WERE CLR. I SHOULD HAVE RECOVERED FROM LNDG INSTEAD OF CONTINUING THE FLARE. THAT WAS THE FAILURE ON MY PART.