Dear, now let us again take a closer look at some of the physical systems in motion ;

a) A simple pendulum with a swinging mass m at the end of a fixed light string of length l.

b) a torsional pendulum disc supported by and swinging about the same suspension wire/string.

c) A mass (m) attached to the free end of a fixed spring moving back and forth on a frictionless floor. 

d) A liquid column moving up and down a U-tube of uniform corss-sectional area about its equilibrium position of equal levels in each limb.

e) An electrical circuit having an inductance L across a capacitance C carrying a charge q.

So, mechanical as well as Electrical systems are now equivalent in their treatment.

A small displacement  from its equilibrium position sets up a restoring force, this force is proportional and acts in such a direction towards the equilibrium position.

The equilibrium or rest position is equivalent to the mean position of a SHO

These observations are very vital to the understanding of the mechanism of S.H.M. 

So, this restoring force is because of elasticity of the given medium. The disturbed system tries to recover and restore its original position after the deforming force is removed. The inertia of the medium comes into the picture now. Because of this property, the motion is repeated on either side of the mean position. In the absence of elasticity the recovery from the disturbed position of the given medium is not possible. And without inertia the undulations are not repeated. Thus we come on conclusion , that elasticity and inertia are two essential properties of a medium to sustain any harmonic wave motion.