All simple harmonic motion systems have these two features:

No loss of mechanical energy.

A restoring force or torque that is proportional to the displacement from equilibrium. The force or torque is also opposite to the displacement - that's what "restoring" implies.

Implications of these facts

The motion of the system is described by an equation of the form
x(t) = A cos(ωt + φ).

The relationship between the acceleration and the displacement is:

a = -ω² x

where ω is the angular frequency of the system.

The period of oscillation is

T

=

2π ω