The 1 Q of a damped harmonic oscillator The damped harmonic oscillator has an equation of motion as follows m xdotdot = - b xdot - k x. This can be rewritten as xdotdot + 2 xdot + o2 x = 0 , where =b/2m, and o = (k/m) . A solution to this equation is x(t) = A exp(-t) cos ( 1t + ) . 1 is the underdamped oscillation frequency (slightly smaller than the undamped frequency o): 12 = o2 - 2 . The velocity of this oscillator is xdot(t) = A exp(-t) [ - 1 sin ( 1t + ) - cos( 1t + ) ] The total energy is E(t) = k x(t)2 + m (dx(t)/dt)2 . This is of the form E(t) = exp(-2t) f(t), where f(t) = m A2 [ ( o2 + 2) cos2( 1t + ) + 2 1 sin( 1t + ) cos( 1t + ) + 12 sin2( 1t + ) ]. f(t) is clearly periodic in t, with a period T = 2/ 1 . The time average of f(t) over one or more cycles in T is <f(t)> = m A2 [ ( o2 + 2) + 12 ] . For lightly-damped oscillators, exp(-2t) varies slowly over a few cycles of T, and A may be regarded as a constant, so we may write <E(t)> = exp(-2t) <f> We will abbreviate <E(t)> to just <E>, and take its time derivative: d<E>/dt = -2 <E> This may be rearranged to - d<E>/dt / <E> = 2 . Eq 3.35, p. 109 rev March 21, 2005
2 <E> represents the oscillator's total energy, averaged over several cycles, and this equation tells us that the fractional decrease in average energy time with equals 2. A quality factor Q is often assigned to lightly damped oscillator, where Q is the ratio of stored energy in the oscillator to the energy dissipated per radian Q = E/[ -dE/d ] When dE/d is written as (dE/dt)/(d/dt) we have Q = E/[ -dE/dt / d/dt ] .
Since dE/dt is P, the power dissipated, and d/dt is the angular frequency , we have Q = E/ [ -dE/dt ] = E/P = (stored energy/ power dissipated) . When we regard E in this expression as the time-averaged energy <E> we may use -dE/dt / E = 2 from above and write Q = /(2) . Eq. 3.64, p. 121
This permits us to calculate the quality factor Q of an oscillator when we know its angular frequency and the average rate at which its energy decreases. If we have the amplitude decay curve for a lightly-damped oscillator, we can fit a decaying exp...