Simple harmonic motion
There is a displacement from the midpoint
There is a force which act in the opposite direction to the displacement which acts towards the midpoint and pushing the object back towards the midpoint
The force is making the object accelerate towards the midpoint
Hence the definition for simple harmonic motion is that it is an oscillation where acceleration is directly proportional to the displacement from the midpoint and directed to the midpoint.

The displacement , velocity and acceleration are all vector quantities therefore the direction always counts.
When the object is at maximum negative displacement, velocity is going to be zero- remember back to m1 where at the highest point the velocity is zero. At the highest negative displacement, the force is in the opposite direction hence there is maximum acceleration in the opposite direction. Therefore negative displacement means positive acceleration. When the object is going back to the equilibrium position, the velocity is going to be maximum. At equilibrium position, the object is going back to where it started hence displacement is zero. Moreover there is no resultant force hence acceleration is zero.

the gradient of the displacement graph will equal velocity the gradient of the velocity graph will equal acceleration

It's also important to note that for SHM, the time period of the oscillations is constant and doesn't change even if the amplitude is changing.
Look in the small book to see that the maximum value for velocity and for acceleration are aw and aw2
Energy gets transferred between kinetic energy and gravational potential energy
The sum of the two stays constant
No energy is lost form the oscillation unlessed it is dmaped
The value for the keitnc energy is ½ mv2
For the gpe if it is something like mass on a pendulum then use mgh wheresas for a spirnf is 1/2kx squared where k is the stiffness constant.
Free vibrations – No transfer of energy to from the surroundings- if a mass is left to osciallte, it will at its nautral frwuqnecy with the same amplitude forever but in reality this never happens.
Forced vibrations happen when there is an