PHYS 422 Lecture 18 Coupled Oscillations and Normal Modes I

# PHYS 422 Lecture 18 Coupled Oscillations and Normal Modes I...

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l d O ill ti d Coupled Oscillations and ormal Modes I Normal Modes I

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Coupled Pendulums so far we have restricted ourselves to systems with one type of vibration characterized by a single natural frequency ω 0 real physical systems may have multiple modes of vibration with many different resonant frequencies x B x A
Coupled Pendulums let’s consider our prototype system in the following configuration spring is at its laxed length relaxed length no force on A or B x B x A s C t each pendulum oscillates at its natural frequency ω 0 ( ω 0 2 = g/ l ) 0 0 cos cos A B xC x Ct ω = = normal mode of the system

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Coupled Pendulums now let’s put our prototype into the following state coupling spring is stretched half a cycle later it is compressed x B x A exerts forces on both A and B if the pendula were free restoring force m ω 2 0 x spring is stretched (compressed) distance 2x 2kx 2 2 0 2 20 A AA dx mm x k x dt ω ++ = 2 22 2 0 2 ( 2 ) 0 A cA c k x dt m ωω + += =
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## This note was uploaded on 03/23/2011 for the course PHYS 422 taught by Professor Staff during the Spring '08 term at Purdue.

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PHYS 422 Lecture 18 Coupled Oscillations and Normal Modes I...

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