On August 2, 2008, at 1200 eastern daylight time, a Piper PA-28R-200, N56064, registered to Pine Ledge Aviation LLC, operating as a Title 14 Code of Federal Regulations (CFR) Part 91 personal flight, collided with the ground during a forced landing in the vicinity of Richmond Airport (08R), West Kingston, Rhode Island. Visual meteorological conditions prevailed and no flight plan was filed. The airplane received substantial structural damage to the left wing. The private pilot and three passengers reported no injuries. The flight departed North Central State Airport (SFZ), Pawtucket, Rhode Island, on August 2, 2008, at 1030. The pilot stated he was in cruise flight at 3,000 feet in the vicinity of Richmond, Rhode Island, when he noted an airframe vibration. He pushed the throttle full forward. The manifold pressure and engine rpm increased, and the vibration increased. He then leaned the mixture, and the vibration seemed to decrease. The airplane would not climb, and started a descent. The pilot selected the nearest airport on his global positioning system which was 08R. He proceeded to the airport to make a precautionary landing. Upon arrival at 08R he initiated a visual approach to runway 11. The pilot stated his attention was diverted to maintaining as much engine power as possible on the approach, he realized he was high and fast, and made a go-around. He remained in left closed traffic and was looking for available forced landing areas in the event he needed to make an off airport landing. The pilot turned on final approach and decided to abort the approach. He stated it was a very hot day, with a high density altitude, and the runway was very short. The pilot increased power to gain altitude, and leaned the mixture which resulted in an "alarming" decrease in airspeed. The airplane gained some altitude, he observed an open sod field, and made the decision to land. Upon reaching the end of a tree line the pilot observed what appeared to be a dirt road in the sod field. He continued toward the field and decided to make a gear up landing. He cut the throttle, lowered the flaps between 10 to 20 degrees and initiated the approach for landing. The pilot stated that he failed to latch the automatic gear lever in the override position to prevent the landing gear from extending when he reduced power during the landing flare. The gear extender warning horn sounded, the gear extender warning lights illuminated and "he ignored the warning horn and the gear extender lights in order to continue his concentration on landing the airplane." The airplane collided with the ground, skidded about a 100 feet, made a left turn, and came to a stop. Review of airport information for 08R, revealed the airport is at an elevation of 130 feet. Runway 11 is 2,129 feet long and 30 feet wide. The temperature at the time of the accident was 30-degrees Celsius, with a density altitude of 1, 960 feet. Review of the Piper PA-28R-200 Pilot's Operating Handbook, Landing Distance versus Density Altitude Chart states the airplane will require 795 feet of runway to land. The Pilot's Operating Handbook states on page 4-4, GEAR UP LANDING, "On aircraft equipped with the automatic gear extender the mechanism will extend the gear below approximately 105 MPH IAS with power off. Be prepared to latch the emergency override lever UP before airspeed drops to 115 mph to prevent landing gear from inadvertently free falling, unless gear extension is desired." Examination of the engine assembly by a Federal Aviation Administration (FAA) inspector, revealed the No. 1 cylinder exhaust valve pushrod would only open .125 inch. The No. 1 cylinder exhibited "heavy carbon build up (running rich) on upper and lower spark plugs. The cylinder exhaust valve and seat, exhaust valve pushrods, and exhaust valve hydraulic lifters were bore scoped and no anomalies were noted. An engine compression check was completed and compression and suction was obtained on all cylinders. The propeller was rotated by hand and continuity of the valve and drive train was confirmed. The left magneto was examined and the rotor and distributor were worn. The right magneto was examined and the rotor and distributor were new. The engine air filter exhibited a "smoky carbon/oil residue." The FAA inspector noted, "in this inspector's opinion, the most probable cause for the engine running rough and airframe vibration as reported is due to the number one cylinder was not performing as expected. The heavy carbon build-up and exhaust valve displacing only .125 of inch."