Mechanical math

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Relative motion of two particles (Theorem 1.2, pg 2l)

In-phase - if the normal mode displacement ratio is positive Phase-opposed * if the normal mode displacement ratio is

negative

1

O', -trr

Hr H,

m1

H, lIu

Particles displaged from equilibrium (pS 25)

F*-) ot xt

m, +-+)

Here, we consider changes in spring forces AII, due to displacements of particles from their equilibrium positions.

The changes in spring forces for the above configuration ffo, AH1 : -ktr1i AH2 : - k2 (x, - xr) (-l) : kz (xz - x) i AH3: -AH2: -kz(xz-x)i AII4 : - fr: (0 - xz) (-D : - kt xzi

Applying Newton's 2"d Law, the equations of motion ffio, ffir*ri = AHr * AHz - - kr h L+ kz @z- x) i nz *zl= AIIr + AII4 - - kz (xz- x) i - kt xzi

Resolving in the j -direction gives, ffir*r=-(h+k) rr * kzxz ffiz iz* - k2 x1* (kz + kixz

PagP 9