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17.3: Normal Modes

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The physical motion corresponding to the amplitudes eigenvector (1,1) has two constants of integration (amplitude and phase), often written in terms of a single complex number, that is,

(θ1(t)θ2(t))=(11)ReBeiω0t=(Acos(ω0t+δ)Acos(ω0t+δ)),B=Aeiδ

with A,δ real. 

Clearly, this is the mode in which the two pendulums are in sync, oscillating at their natural frequency, with the spring playing no role.

In physics, this mathematical eigenstate of the matrix is called a normal mode of oscillation. In a normal mode, all parts of the system oscillate at a single frequency, given by the eigenvalue.

The other normal mode,

(θ1(t)θ2(t))=(11)ReBeiωt=(Acos(ωt+δ)Acos(ωt+δ)),B=Aeiδ

where we have written ω=ω20+2k. Here the system is oscillating with the single frequency ω, the pendulums are now exactly out of phase, so when they part the spring pulls them back to the center, thereby increasing the system oscillation frequency.

The matrix structure can be clarified by separating out the spring contribution:

M=(ω20+kkkω20+k)=ω20(1001)+k(1111)

All vectors are eigenvectors of the identity, of course, so the first matrix just contributes ω20 to the eigenvalue. The second matrix is easily found to have eigenvalues are 0,2, and eigenstates (1,1) and (1,−1).


This page titled 17.3: Normal Modes is shared under a not declared license and was authored, remixed, and/or curated by Michael Fowler.

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