The airplane was parked on the operator's ramp when Federal Aviation Administration Inspector(s), on a routine ramp inspection, found that the airplane had substantial damage to the firewall and structure aft of the firewall. Airplane maintenance records showed an entry stating that the damage was due to a suspected hard landing. The operator stated that the damage must have occurred between the airplane's 100-hour inspection and oil change performed September 27 and October 24, 2016, respectively. The operator's mechanic stated the damage was found when the engine cowl was removed for the oil change. A review of airplane flight logs showed approximately 20 different people had flown the airplane over 48 flights between the time of the 100-hour inspection and the oil change. On October 8, 2016, a student pilot was performing touch and go's when the airplane bounced 2 or 3 times during the flight's final landing. The student pilot stated that when she turned off the runway onto the taxiway, she noticed the airplane was not steering properly and reported to air traffic control tower that she had some mechanical difficulty that required maintenance assistance. The student pilot's flight instructor and a company lineman responded and noticed that the airplane had a flat nose wheel tire. The lineman inflated the tire, and the student pilot and flight instructor taxied the airplane to the operator's maintenance facility. The flight instructor informed the operator's mechanic of the bounced landing and flat tire. The mechanic stated he replaced the nose tire, performed a brief exterior walk-around visual inspection of the aircraft, noting it appeared airworthy and returned it to service. A hard landing inspection was not performed. Other than the bounced landing reported on October 8, 2016, there were no other operator concerns of the 48 separate flights conducted between inspections.