Problem Set-1 c Week 11Challenge : Transient Handling

Given parameters:-

undamped natural frequency,Wn=[YB*NR+NB*M*U-yR*NB]/[M*U*Izz)
here, Yb=Cf+Cr
Nr=(a^2*Cf+b^2*Cr)/u
Nb=(a*Cf-b*Cr)
yR=(a*Cf-b*Cr)/u

a) Passenger car
1) undamped natural frequency
yb=13752+11460=25212lbs/rad
Nr=(2.92^2*13752 +5.41^2*11460)/60=7544.45lbs-ft-sec/rad
Nb=2.92*13752-5.41*11460=21842.76lbs-ft/rad
yR=21842.76.60=364.046lbs-sec/rad
taking mass M in slugs,M=2000*0.453592/14.594=62.16sluf-ft^2
Izz=(1128/14.594)*11.11 = 832 slig-ft^2
Wn^2=[(-25212*-7544.458)+(21842.76*62.16*60)-(-364.04*21842.76)]/[62.16*60*832]
=87.66(rad/sec)^2
Wn=9.36rad/sec = 9.36/6.28 = 1.49044 Hz

2) Damping ratio
=C/C_critical
=C/2sqrt(k*Izz)
=C/2*Izz*Wn
and C=-[Nr+(Izz*Yb)/(M*u]=13173lbs-s/ft
C_critical=2*Izz*Wn=15575.04lbs-s/ft

damping ratio = 15575.04/13173=0.8453

3) Damped frequency calculation
W=Wn*sqrt(1-C^2)
=9.36*sqrt(1-0.8453^2)
=5.0054rad/sec
=0.797Hz