The differential equation for the constrained center of grav

The differential equation for the constrained center of gravity pitching motion of an airplane is computed to be alpha.. + 4 alpha. + 36 alpha = 0 Find the following: omega_n, natural frequency, rad/s zeta, damping ratio omega_d, damped natural frequency, rad/s

Solution

Comparing the equation with standard spring-mass-damper equation mx\'\' + cx\' + kx = 0 we get,

m = 1

c = 4

k = 36

a)

Natural frequency w_n = (k/m)^0.5

= (36/1)^0.5

= 6 rad/s

b)

Damping ratio z = c / (2 (mk)^0.5)

= 4 / (2*(1*36)^0.5)

= 0.333

c)

Damped Natural frequency w_d = w_n* (1-z²)^0.5

= 6 * (1 - 0.333²)^0.5

= 5.657 rad/s