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PHYS 422 Lecture 14 Free Vibrations III

# PHYS 422 Lecture 14 Free Vibrations III - Free Vibrations...

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Free Vibrations f Physical Systems III of Physical Systems III

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Damping in Oscillators if we charted the amplitude as a function of time we would et get the resistive force of a fluid to a moving object is given by 2 12 () R vb v b v =+ magnitude of the velocity as long as v is small compared to b 1 /b 2 we can eglect the v 2 rm neglect the v term Newton II is 2 x 2 dx mk x b v dt = −−
Damping in Oscillators 2 2 dx mk x b v dt = −− e differential equation becomes the differential equation becomes 2 2 0 d x mb k x t dt + += dt 2 22 0 0 d x b k x γ ω ⇒+ + = = = 00 2 dt d tm m γγ describes damping of the system gular frequency if ndamped angular frequency if undamped let’s assume an exponential solution where x presents the real part of a rotating vector z represents the real part of a rotating vector z satisfying our differential equation 2 ( ) t z d z 2( 0 2 with i pt dz z zA e dt dt α + ++ = =

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Damping in Oscillators 2 2( ) w i t h ip t dz d z z A e α + + = = 0 2 0 with zz dt dt γω ++ substituting z into our differential equation we get ( ) 22 ( ) 0 0 t pi p A e + −+ + =
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PHYS 422 Lecture 14 Free Vibrations III - Free Vibrations...

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