The pilot reported that he made a normal landing on the main tires, but when the nose wheel set down it began to shimmy violently. In response, he pulled back on the control yoke and the airplane became momentarily airborne before it bounced several times and porpoised down the runway before stopping. Postaccident examination of the airplane revealed the nose wheel tire was crushed and partially attached to the rim; the firewall was wrinkled, the nose gear landing doors were damaged, the nose gear strut was slightly bent, and all three propeller blades were bent aft at the tips. Further examination of the runway indicated there were two propeller strike marks on the runway before the nosewheel struck the ground and left its own impact scars. Just forward of where the nosewheel impact marks began, were four additional propeller strike marks. The pilot stated that he did not observe any abnormality with the nosewheel tire prior to the flight.