Solution

Seek

z = v e^{i Ω t}

and put

μ = m / M

. The generalized characteristic equation is

det (K - Ω^{2} M) = 0,

(1 + μ) (ω_{0}^{2} - Ω^{2})^{2} - μ Ω^{4} = 0.

As a quadratic in

X = Ω^{2}

, this is

X^{2} - 2 (1 + μ) ω_{0}^{2} X + (1 + μ) ω_{0}^{4} = 0,

Ω_{\pm}^{2} = ω_{0}^{2} (1 + μ \pm μ (1 + μ)) .

A corresponding displacement vector is

v_{\pm} = (ω_{0}^{2} - Ω_{\pm}^{2} Ω_{\pm}^{2}) .

Thus the four normal modes are

v_{+} e^{\pm i Ω_{+} t}

and

v_{-} e^{\pm i Ω_{-} t}

. Both squared frequencies are positive for every

μ > 0

, proving linear stability.

For

μ ≪ 1

Ω_{\pm} = ω_{0} (1 \pm \frac{1}{2} μ + O (μ)) .

Therefore, labeling the larger and smaller positive frequencies by

ω

and

ω^{'}

ω - ω^{'} = ω_{0} μ + O (μ),

α = ω_{0}

Solved by gpt-5.6-sol high.

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