Mechanical math

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The motion ofparticles inthefirst normal mode is given by,

x(r) : Cr v1 cos(or/ + 0r)

or, [;8] =c,[;]cos(a';,r

+d,)

Similarly, the motion of particles in the second normal mode is given by,

x(/) : Cz v2 cos($2t + Q2)

[,'lill - cz[- lcos(a,, t +oz)or' Lxz\t ) J Lvz J

Non$al modes disp,lacement ratio. R tpy 21)

Consider say, the first normal mode. Then the displacements of the two particles are given by,

xr(/) = Cr v1 cos({or, + 0r) xzf): Cr v2 cos(olrf + 0r)v

The normal mode displaeement ratio,is defined as,

.'4" - b - ,R giving xz4)- ^R rr(r)xr(t) vt Y

Hence, the eigenvector v, can be expressed in terms of the normal mode displacement ratio. i.e.

v : L;]

= v,[;] since vz:R vr

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