Conservation of mechanical energy applies to a simple harmonic motion system. We should find that the sum of the potential and kinetic energies is constant.
U = ½ kx2 = ½ kA2 cos2(ωt + φ)
K = ½ mv2 = ½ mA2ω2 sin2(ωt + φ)
We know that ω2 = k/m, so the kinetic energy expression becomes:
K = ½ kA2 sin2(ωt + φ)
Because sin2(something) + cos2(something) = 1
U + K = ½ kA2 = constant
Also, U + K = ½ mA2 ω2
Graphs of potential and kinetic energy as a function of time show that the total energy is constant, and that energies go through two complete cycles for each oscillation of the object.
Graphing the energies as a function of position is also interesting.