GATE 2021 AE – Question 42
A two degree of freedom spring-mass system has natural frequencies 233.9 and 324.5 rad/s and corresponding mode shapes φ1=(1,−3.16), φ2=(1,3.16). With the following initial displacements in cm and zero initial velocities, which statements are true?
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: (B) (2,−6.32) excites only the first mode.; (C) (2,−2) excites both modes.
Explanation
For a two-degree-of-freedom system, the free response is a combination of the two modes:
$$\mathbf x(t)=c_1\boldsymbol\phi_1\cos(\omega_1t+\psi_1)+c_2\boldsymbol\phi_2\cos(\omega_2t+\psi_2).$$
With zero initial velocities, a given initial displacement $\mathbf x_0=c_1\boldsymbol\phi_1+c_2\boldsymbol\phi_2$ excites **only one mode** if $\mathbf x_0$ is a multiple of that mode shape. Here $\boldsymbol\phi_1=(1,-3.16)$ and $\boldsymbol\phi_2=(1,3.16)$.
- **A. (6.32, −3.16):** not a multiple of either shape. The ratio is $-0.5$, but the mode ratios are $\mp3.16$. Excites both. ✗
- **B. (2, −6.32):** $=2\times(1,-3.16)=2\boldsymbol\phi_1$. Only the first mode. ✓
- **C. (2, −2):** ratio $-1$, which is not $\pm3.16$, so it is a mix of both modes. Excites both. ✓
- **D. (1, −6.32):** ratio $-6.32$, not $-3.16$. Excites both. ✗
Answer **B and C**.