Damped Harmonic Mean, Physics Help
md2x /dt2 + r dx / dt + Kx = 0
Or d2 x / dt2 + r/m dx / dt + k / m x = 0
Or d2 x / dt2 + 2b dx / dt + ω2 x = 0
B = r / 2m is called damping coefficient
Solution to the equation is
X = x0 / 2 e –bt [(1 + b / b2 –ω2) e +1 √(b2– ω)2 + ( 1 – b / b2 – ω2) e –t √(b2-w2) ]
Note that x0 = x0e –bt is the amplitude at any time t.
If r/2m > √(K/m) motion is non-oscillatory and over damped
If r / 2m = √(K/m) motion is critically damped.
If r/2m =< √(K/m) damped oscillatory motion results.
If r = 0 undamped oscillations result.
Free or natural or fundamental frequency
Forced (c) resonant (d) damped
Free or natural vibrations depend upon dimensions and nature of the material (elastic constants).
If a periodic force of frequency other than the material’s natural frequency is applied then forced vibrations result. For example, if y = y0 sin ωt was the equation of SHM of a particle and a periodic force p sin ω1t if applied then ω ≠ ω1 then, y = y0 sin ωt + p sin ω1t.
The resultant frequency is different from the natural frequenc of oscillation.
Resonant oscillations are a certain type of forced vibrations. If frequency of applied force is equal to the natural frequency of the source
That is y = y0 sin ωt + p sin ωt = (y0 + p) sin ωt
That is amplitude increases or intensity increases with resonance.
In damped oscillations amplitude of the vibration falls with time as shown.
Amplitude at any instant is given by
Y = y0 e – bt
Where y0 amplitude of first vibrations and y is is amplitude at time t and b is damping coefficient.