The two-degree-of-freedom system shown has initial displacements x₁ = 2 cm and x₂ = 1.24 cm and vibrates in its first mode. If its first mode shape is $\phi_1=[1\ a]^T$, find a to 2 decimals.
A pure-mode displacement vector is proportional to its mode shape. Normalize the first component to one: $a=x_2/x_1=1.24/2=0.62$. The positive value indicates in-phase motion. The drawn equal masses and springs are consistent with the first-mode ratio approximately 0.618.
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