Damped Harmonic Mean
md2x /dt2+ r dx / dt + Kx = 0
Or d2x / dt2+ r/m dx / dt + k / m x = 0
Or d2x / dt2+ 2b dx / dt + ω2x = 0
B = r / 2mis 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 thatx0=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 = 0undamped 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, ify =y0sin ωt was the equation ofSHMof a particle and a periodic forcep sin ω1tif applied thenω ≠ ω1then,y =y0sin 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 isy =y0sin ωt + p sin ωt = (y0+ p) sin ωt
这是增加振幅或我ntensity increases with resonance.
In damped oscillations amplitude of the vibration falls with time as shown.
Amplitude at any instant is given by
Y =y0e – bt
Wherey0amplitude of first vibrations andyis is amplitude at timetandbis damping coefficient.
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