A 2-DOF undamped structural system has mass matrix $[M] = \begin{bmatrix} m & 0 \\ 0 & m \end{bmatrix}$ and stiffness matrix $[K] = \begin{bmatrix} 2k & -k \\ -k & 2k \end{bmatrix}$. The ratio of the higher natural frequency $\omega_2$ to the lower natural frequency $\omega_1$ is:

(A) $\sqrt{3} \approx 1.732$

(B) $3.0$

(C) $\sqrt{2} \approx 1.414$

(D) $2.0$

Answer Key:

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### 1. Strategy & Theoretical Principles The characteristic equation for free undamped vibration is: $$\det([K] - \omega^2 [M]) = 0$$ With $[M] = m [I]$ and $[K] = \begin{bmatrix} 2k & -k \\ -k & 2k \end{bmatrix}$: $$\det\begin{bmatrix} 2k - m\omega^2 & -k \\ -k & 2k - m\omega^2 \end{bmatrix} = 0$$ ### 2. Analytical Formulation & Calculation Let $\lambda = m\omega^2$: $$(2k - \lambda)^2 - (-k)^2 = 0$$ $$(2k - \lambda)^2 - k^2 = 0$$ Factor as a difference of squares: $$[(2k - \lambda) - k][(2k - \lambda) + k] = 0$$ $$(k - \lambda)(3k - \lambda) = 0$$ Roots: $$\lambda_1 = k \implies \omega_1^2 = \frac{k}{m} \implies \omega_1 = \sqrt{\frac{k}{m}}$$ $$\lambda_2 = 3k \implies \omega_2^2 = \frac{3k}{m} \implies \omega_2 = \sqrt{\frac{3k}{m}}$$ Ratio of natural frequencies: $$\frac{\omega_2}{\omega_1} = \frac{\sqrt{3k/m}}{\sqrt{k/m}} = \sqrt{3} \approx 1.732$$ ### 3. Distractor Forensics - Candidate squaring the ratio gets 3.0. - Candidate using $\sqrt{2}$ miscalculates cross term. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Distinguish eigenvalue $\lambda = \omega^2$ from natural frequency $\omega$. - **Standard Reference:** Rao, S.S., *Mechanical Vibrations*, Chapter 5 (Two-Degree-of-Freedom Systems).

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