Introduction to Harmonic Motion

Simple Harmonic Motion

Damped Harmonic Oscillators

Under damped systems

Over damped systems

Critically damped systems

Equations of motion for a damped pendulum

Applet

Simple Harmonic Motion

SHM can be applied to many situations in a diverse range of fields ranging from Quantum Mechanics, where vibrational quantum states can be predicted with good accuracy, to automotive suspension, pendula and electric LRC circuits.

The restoring force is described by Hooke’s law, where F is the restoring force, x is the displacement and k is the spring constant

F = -kx(1)

For a time varying force Equation (1) can be re-written as

F(t) = -kx(t)(2)

Physical systems are usually more complicated than a Hooke’s law system. For example a fixed spring with a mass on its other end can be approximated using Hooke’s law, but if appropriate measurements were taken then a damping force proportional to the velocity of the oscillation would be noticed. Similarly an inertial effect would be observed where the inertial force was proportional to the acceleration. Combining these forces results in equation 3

(3)

Where F(t) is the time dependant force, k is the spring constant, b is the damping constant and m is the mass (or inertial constant). For most systems k, b and m will be constants.

Equation (3) is a linear, second-order differential equation. This means that a linear combination of solutions will also be a solution i.e. if x(t) = Q and x(t) = W are solutions then x(t) = Q + W is a solution

It is second order because the differential power is 2, so there are 2 linearly independent solutions

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