Question

The longitudinal motion of an aircraft is governed by the following characteristic equation for its Short Period Mode: \[ s^2+4.8s+16=0. \] Calculate the damping ratio.

• A. \(0.8\)
• B. \(0.6\)
• C. \(0.7\)
• D. \(0.5\)

Hint:

For a second-order system, \[ \boxed{ s^2+2\zeta\omega_n s+\omega_n^2=0 } \] Compare coefficients directly to determine \[ \boxed{\zeta} \] and \[ \boxed{\omega_n.} \]

Answer 1

The Correct Option is B

Step 1: Write the standard second-order characteristic equation. The standard form is \[ s^2+2\zeta\omega_n s+\omega_n^2=0, \] where \[ \zeta=\text{damping ratio}, \] and \[ \omega_n=\text{natural frequency}. \]

Step 2: Compare with the given equation. Given, \[ s^2+4.8s+16=0. \] Comparing, \[ \omega_n^2=16, \] \[ \omega_n=4. \] Also, \[ 2\zeta\omega_n=4.8. \] Hence, \[ 2\zeta(4)=4.8. \] \[ 8\zeta=4.8. \] \[ \zeta=\frac{4.8}{8}=0.6. \] Therefore, \[ \boxed{0.6} \] is the damping ratio. Thus, \[ \boxed{(B)} \] is the correct answer.