The steady-state vibration displacement $y(t)$ of an airframe sensor mount satisfies $\frac{d^2 y}{dt^2} + 9y = 18\cos(3t)$. If the initial conditions are $y(0) = 0$ and $y'(0) = 0$, what is the amplitude coefficient of the resonant term $t\sin(3t)$ in the solution?

Answer Key:

3

### Solution: Homogeneous roots are $\pm 3i$. The excitation $18\cos(3t)$ is at resonance. Particular solution form: $y_p(t) = t(A\cos(3t) + B\sin(3t))$. Substituting into $\ddot{y} + 9y = 18\cos(3t)$ gives $6B\cos(3t) - 6A\sin(3t) = 18\cos(3t) \implies 6B = 18 \implies B = 3$ and $A = 0$. Thus $y_p(t) = 3t\sin(3t)$. Amplitude coefficient is 3. Correct: 3.

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