Can anyone explain why simple harmonic motion is defined by acceleration being proportional to the negative displacement?
Please see below.
There are several different characteristics of harmonic systems. In any harmonic system there is a preferred location. If it is displaced from the preferred location, there will be a force to return it to the preferred location.
Imagine it is displaced away from the preferred location. The displacement can be in either direction from the preferred location -- the force will be such to move the system back to the preferred location.
The relationship between the amount of displacement and the amount of the restoring force varies significantly in types of harmonic systems. Many of the different types of harmonic systems have been studied to determine an equation to describe its motion.
With a common type of harmonic system, the restoring force is directly proportional to the amount of displacement. The formula that describes that is
Think of a mass hanging from a spring. Displacing the mass from the preferred location, in either direction, will cause the spring to develop a force imbalance that is proportional to the displacement. The formula above describes that relationship. That force will move the mass back toward the preferred location.
The motion of this type of harmonic system is called simple harmonic motion because the formula that describes it is the most simple.
I hope this helps,