Q value of an underdamped oscillator (following text treatment of T. L.Chow)

• E = 1/2 m x-dot^2 + 1/2 k x^2
• underdamped oscillator: E(t) = C exp(-beta t) [function f(t) periodic in T = 2 pi/omega]
• easy to show that energy is E(t) = exp(-2 beta t) [ function g(t) periodic in T]
• if we evaluate at intervals of nT where n is an integer we get
• E(nT) = exp(- 2 beta t) E(0)
• if we use the notation <E(t)> for E evaluated at intervals of T we get
• <E(t)> = exp(- 2 beta t) <E(0)>
• this means d<E(t)>/dt = - 2 beta <E(t)>
• T. L. Chow' text defines Q of an underdamped oscillator as
• Q = energy/(magnitude of energy dissipated per radian)
• energy dissipated per radian is energy dissipated in the time for the function to oscillate through one radian
• omega =radians/sec, so 1/omega = seconds/radian
• thus, energy lost per radian = d<E(t)>/dt * 1/ omega, or
• energy lost per radian = -2 beta/omega <E(t)>, and finally that
• Q = <E(t)>/[2 beta/omega <E(t)>] = omega/(2 beta).
• Q = omega/(2 beta). [ T. L. Chow]
• omega^2 = sqrt(omega-0^2 + beta^2)
• Marion and thornton have Q = [omega at resonance]/(2 beta), slightly different from Chow
• Only for heavy damping will there be any difference between the two statements.

Q by measuring the frequency difference between the 0.707 points

• The response function to a sinusoidal driving function is
• A/sqrt( a^2 + b^2), where a = omegao^2-omega^2 and b = 2 beta omega
• at the max of the response a is nearly zero and omega=omegao
• so max response = A/b
• at the 0.707 response points a = b, so
• (omegao+omega)(omegao-omega) = plus or minus 2 beta omega
• since omega is nearly equal to omegao we have that
• omegao-omega = beta at the 0.707 points
• the frequency difference between the 0.707 points is then
• delta-omega = 2 beta
• The Q value is given by omega/(2 beta), so we may also write
• Q = omega/delta-omega where delta-omega is the frequency difference between 0.707 points