Damped Harmonic Mean, Physics Help

**md ^{2}x /dt^{2} + r dx / dt + Kx = 0 **

**Or d ^{2} x / dt^{2} + r/m dx / dt + k / m x = 0**

**Or d ^{2} x / dt^{2} + 2b dx / dt + ω^{2 }x = 0**

**B = r / 2m** is called damping coefficient

Solution to the equation is

X = x_{0} / 2 e –bt [(1 + b / b^{2} –ω^{2}) ^{e +1} √(b^{2}– ω)^{2} + ( 1 – b / b^{2 }– ω^{2}) e –t √(b^{2}-w^{2}) ]

Note that **x _{0}** =

**x**e –bt is the amplitude at any time t.

_{0}**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 =** **y _{0}** sin ωt was the equation of

**SHM**of a particle and a periodic force

**p sin ω**if applied then

_{1}t**ω ≠ ω**

_{1}then,

**y =**

**y**

_{0}**sin ωt + p sin ω**

_{1}t.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 = y_{0} sin ωt + p sin ωt = (y_{0} + 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 = y_{0} e – bt**

Where **y _{0}** amplitude of first vibrations and

**y**is is amplitude at time

**t**and

**b**is damping coefficient.