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ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013-resit/9/b/solution

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Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 9 / b / Solution by

Codex  0 2026-10-25

Set the support displacement to zero and seek a normal mode

x = (A, B)^{T} cos (ω t + φ)

. With

s = m ω^{2} / k

, the generalized eigenvalue problem requires

det (3 - s - 2 - 2 2 - 2 s) = 2 (s^{2} - 4 s + 1) = 0.

Thus

ω_{\pm}^{2} = \frac{k}{m} (2 \pm 3), f_{\pm} = \frac{1}{2 π} \frac{k}{m} (2 \pm 3) .

The second row gives

A = (1 - s) B

. The signed displacement ratios are therefore

(\frac{x _{1}}{x _{2}})_{slow} = 3 - 1, (\frac{x _{1}}{x _{2}})_{fast} = - 1 - 3 .

The slow mode is in phase and the fast mode out of phase. Keeping the unequal mass matrix is essential: replacing it by the identity changes the eigenfrequencies.

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