Question

For a general aviation airplane, one of the complex conjugate pair of eigenvalues for longitudinal dynamics is given by $\lambda = -0.039 \pm 0.0567 i$ (in SI units). If the system is disturbed to excite only this mode, the time taken for the amplitude of response to become half in magnitude is ............ s (rounded off to 1 decimal place).

Hint:

The decay rate of oscillatory modes is governed by the real part of eigenvalues. The half-life is simply $\ln 2 / |\sigma|$.

Answer 1

Step 1: General solution form.
For a mode with eigenvalue $\lambda = \sigma \pm i\omega_d$: \[ x(t) = e^{\sigma t} \big( C_1 \cos \omega_d t + C_2 \sin \omega_d t \big) \] Here, $\sigma = -0.039$ and $\omega_d = 0.0567$. Step 2: Decay factor.
Amplitude decays exponentially with $e^{\sigma t}$. If initial amplitude = 1, then after time $t_{1/2}$: \[ e^{\sigma t_{1/2}} = \frac{1}{2} \] Step 3: Solve for $t_{1/2$.}
\[ \sigma t_{1/2} = \ln \left(\frac{1}{2}\right) = -\ln 2 \] \[ t_{1/2} = \frac{-\ln 2}{\sigma} \] Substitute $\sigma = -0.039$: \[ t_{1/2} = \frac{0.693}{0.039} \approx 17.77 \, \text{s} \] \[ \boxed{17.8 \, \text{s}} \]