Two masses, and , sit between fixed walls, joined in the order wall, spring, ,…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Coupled oscillators, normal modes and Fourier analysis

Modes, Packets and Media, Ch Normal modes of many coupled oscillators: eigenvalues, symmetry and response

Problem

Two masses, mm and 3m3m, sit between fixed walls, joined in the order wall, spring, mm, spring, 3m3m, spring, wall by three identical springs κ\kappa. Its mode frequencies are ωn2=snκ/m\omega_n^2=s_n\kappa/m with s1=(4−7)/3=0.4514s_1=(4-\sqrt7)/3=0.4514 and s2=(4+7)/3=2.2153s_2=(4+\sqrt7)/3=2.2153, and its mode shapes, scaled so that the first component is 1, are (1, 1.549)(1,\,1.549) and (1, −0.215)(1,\,-0.215). The system is released from rest with x1(0)=dx_1(0)=d, x2(0)=0x_2(0)=0. ((a)) Using mass-weighted orthogonality, find the fraction of the total energy that ends up in each mode. ((b)) Check your fractions against the total initial potential energy computed directly from K\mathbf{K}.

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