A knowledgeable aerospace student selects a pressure transdu

A knowledgeable aerospace student selects a pressure transducer (with omega_n = 0281 rad/s and eta = 2.0) to investigate the pressure fluctuations within a laminar separation bubble on the suction side of an airfoil. Assume that the transducer behaves as an over-damped second-order system. If the experiment requires that the transducer response has M (omega) > 0.707 and |phi (omega)|

Solution

solution:

1) here for given overdamped system following terms are

wn=6284 rad/s

zeta=n=2

2) for second order system transfer function is given by

M(s)=wn^2/(s^2+2*n*wn+wn^2)

3) where in vector form it is given as for overdamped systema as

M(jw)=(1/((1-(w/wn)^2)^2+4*n^2*w^2/wn^2)^.5)

and angle is given as

phi=a=tan^-1(2*n/(w/wn-wn/w))

4) as for given system magnitude should be grater>.707

from above equation for magnitude M=.707,we get

wo=1675.2236 rad/s

for angle a<20,from above relation we get maximum frequency supported as on soving

w1=69627.72 rad/s and w2=-576.6 rad/s

as w1>wo,hence maximumfrequency satisfiyng both criteria is

w1=69627.72 rad/s

hence in hertz is

1 Hz=2*pi rad/s

w1=11081.59 Hz