Simple Harmonic Motion (SMH) is derived from Newton’s force law, F = ma and Hooke’s spring law, F = -kx. If we start with -kx = ma, and ask ourselves what function could represent x, we have a differential equation, which is solved by x = A cos ωt. This leads to ω=sqrt(k/m).
We can also use e^iωt as a solution, or at least the real part of it. But these equations do not accurately depict what is observed. When watching an oscillation, eventually it slows down. This is because of the friction due to force, expressed as F = -βv. Now we rewrite the sum of the forces as ma = -kx – βv, and try to find a solution for this differential equation. We guess a solution based on e^iωt, and then solve for ω using the quadratic equation, and plug into the e^iωt equation.
We end up with a dampening constant γ=β/m, and we use this to determine a halflife or 1/e which is about 1/3 the decay time.
An out of phase driver means a small driver can produce large oscillations, leading to Resonance!