At low engine speeds, predominant noises are valve train noises that occur at the instant the valve opens and closes, and those that occur while the follower is on the cam lobe. Valve noises are caused by impacts, and their sound intensity is proportional to the impact speed. Follower noises are caused by frictional vibrations due to metal-to-metal contact at points where oil film thickness becomes zero. Irregular contact surfaces increase these noises. Valve train noises at high engine speeds are caused by irregular valve behavior. The profile of the new cam design is expressed by “n” th-order Fourier series in 360 deg continuously, and has no high-order harmonics that resonate with the natural frequency of the train. Its wide parameter permits design of a most reasonable acceleration curve. Valve motion is thus improved and noise is reduced.