## Simple Harmonic Oscillator

Any spring that obeys Hooke’s Law is appropriately named a “simple harmonic oscillator”. This law of physics describes the behavior of a mass-spring system disturbed from its equilibrium by pulling or pushing on the mass in such a way that the mass experiences a “restoring force” described by Hooke’s Law, F = -sx where “F” is the restoring force, “s” is the spring constant (units – Newtons/meter) and “x” is the displacement of the mass. If x is positive, this refers to stretching the spring, and if x is negative this refers to compression. The spring constant determines how easily the spring is deformed (also known as stiffness). The negative sign refers to the fact that the force opposes the force applied (it is a restoring force), which aligns with Newton’s Third Law of Motion.

Another important Newtonian equation is F = ma which states that force is proportional to both an object’s mass and acceleration. If these two equations are brought together, using the relation that acceleration is the second derivative of position, a second order linear differential equation can be formed.

It is seen that the solution to this undamped mechanical system is sinusoidal in nature. This makes intuitive sense, because sinusoids are proportional to their second derivative, meaning if a sine or cosine is plugged into the differential equation, it can be shown to be a solution. The system’s “natural frequency” is obtained by equating the potential and kinetic energy of the system. This is intuitively satisfying because the kinetic energy dominates the system at frequencies below the resonance/undamped natural frequency and above this frequency, potential energy dominates.

For damped system, the equation becomes slightly more complex as a new force must be considered (the product of mechanical resistance and acceleration). This is a much more realistic approach because in a real mass spring system, air creates a frictional force effect on the spring as it oscillates, causing the oscillations to die out as mechanical/motional energy is lost to heat. The solution to this new differential equation contains a decaying exponential term.

Three cases of damping are shown above: critical damping, overdamping and underdamping. These cases depend upon whether the resonance/undamped natural frequency of the system is equal to the real part of gamma (called temporal absorption coefficient) in which case the system would be critically damped. An underdamped system would occur when the absorption coefficient is lower than the resonance frequency and the opposite would be considered an overdamped system. As shown above, critically damped systems have no oscillations because the imaginary part of gamma is zero (no reactance). Underdamped systems experiences oscillations that decay to zero over time. Overdamped systems decay to equilibrium without oscillating, but not as quickly as critically damped systems.

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