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Problem 6.59PP
Golden Nugget Airlines had great success with their free bar near the tail of the airplane. (See 
Problem) However, when they purchased a much larger airplane to handle the passenger 
demand, they discovered that there was some flexibility in the fuseiage that caused a iot of 
unpleasant yawing motion at the rear of the airplane when in turbulence, which caused the 
revelers to spill their drinks. The approximate transfer function for the rigid body roll/yawl motion, 
called the "Dutch roll” mode (Section 10.3.1) is 
r(s) 8 .75(4 j ^ + 0 .4 s + 1)
M s ) ^ (s /0 .0 1 + l ) ( s 2 + 0 . 2 4 S + 1 ) ’
where r is the airplane’s yaw rate and dr is the rudder angle. In performing a finite element 
analysis (FEA) of the fuselage stmcture and adding those dynamics to the Dutch roll motion, they 
found that the transfer function needed additional terms which reflected the fuselage lateral 
bending that occurred due to excitation from the rudder and turbulence. The revised transfer 
function is
found that the transfer function needed additional terms which reflected the fuselage lateral 
bending that occurred due to excitation from the rudder and turbulence. The revised transfer 
function is
r(s) 8.75(4^2+0.4s+ 1)
M s ) (s/0.01 + I)(s2 + 0 .2 4 s + 1) + 2{s/o)i, + 1) ’
where rub is the frequency of the bending mode (= 10 rad/sec) and ^ is the bending mode 
damping ratio ( =0.02). Most swept-wing airplanes have a “yaw damper," which essentially feeds 
back yaw rate measured by a rate gyro to the rudder with a simple proportional control law. For 
the new Golden Nugget airplane, the proportional feedback gain /( = 1, where 
Eq.1
5r(s) = -Kr(s).
(a) Make a Bode plot of the open-loop system, determine the PM and GM for the nominal design, 
and plot the step response and Bode magnitude of the closed-loop system. What is the 
frequency of the lightly damped mode that is causing the difficulty?
(b) Investigate remedies to quiet down the oscillations, but maintain the same low-frequency gain 
in order not to affect the quality of the Dutch roll damping provided by the yawrate feedback. 
Specifically, invesfigate each of the following, one at a time;
(i) Increasing the damping of the bending mode from ^ = 0.02 to ^ = 0.04 (would require adding 
energy-absorbing material in the fuselage structure).
(ii) Increasing the frequency of the bending mode from rub = 10 rad/sec to rub = 20 rad/sec 
(would require stronger and heavier structural elements).
(iii) Adding a low-pass filter in the feedback—that is, replacing A in Eq. (1) with KDc (s), where
Dc(s) =
s /r p -p 1
(iv) Adding a notch filter as described in Section 5.4.3. Pick the frequency of the notch zero to be 
at tub, with a damping of ^ = 0.04, and pick the denominator poles to be (s /100 + 1) 2, keeping 
the DC gain of the filter = 1.
(c) Investigate the sensitivity of the preceding two compensated designs (iii and iv) by 
determining the effect of a reduction in the bending mode frequency of-10%. Specifically, 
reexamine the two designs by tabulating the GM,PM, closed-loop bending mode damping ratio, 
and resonant-peak amplitude, and qualitatively describe the differences in fhe step response.
(d) What do you recommend to Golden Nugget to help their customers quit spilling their drinks? 
(Telling them to get back in their seats is not an acceptable answer for this problem! Make the 
recommendation in terms of improvemenfs to the yaw damper.)
Problem
For the open-loop system
AT(»-f 1)KG(s) =
j2(j -F10)2’
determine the value for K af fhe stability boundary and the values of K af fhe points where PM = 
30”.
Step-by-step solution
S le p t o f21
The transfer function is given by 
r ( s ) _ 8 . 7 5 ( 4 s ^ -f 0 .4 s -f 1)
( 3 ^ + l) (4 ’ + 0.24a + l ) ’
■Where,
r is the airplane’s yaw rate and is the rudder angle.
Step 2 of 21
The MATLAB program to draw the bode plot for the given open loop function ii 
» num =8.75*[4 0 .4 1 ] :
» f l= [ l / D .D l 1] ;
» f 2 = [ l 0 .2 4 1] ;
» d e n = c o n v (f l , f 2 ) ;
» sy s= tf(n u m , den)
» bode (sys)
» g r id
Step 3 of 21 -e.
The Bode plot obtained onexecutionofthe program is 
B ede D iagram
Step 4 of 21
The gain and phase margins from the Bode plot are 
|gj|/=oo|
\PM = ?7.5° at ai = 0.085 rad^
Step 5 of 21
The revised system is 
r (s ) 8.75(4s^+ 0 .4 s -F l)
Where,
Ifs’ -F 0.24s+
= 1 0 rad/sec 
^■=0.02
The MATLAB code to plot the Bode plot for the revised open loop system
IS
» n u m = 0 .7 5 * [ 4 0 . 4 1 ] ;
» f l = [ l / 0 . 0 1 1 ] ;
» f 2 = [ l 0 . 2 4 1 ] ;
» f 3 = [ l / 1 0 0 0 . 0 4 / 1 0 1 ] ;
» d e n = c o n v ( f l , c o n v ( f 2 , f 3 ) ) ; 
» s y s = t f ( n u m , d en )
» m a r g i n (s y s )
» g r i d
Step 6 of 21 -o.
The Bode plot obtained onexecutionofthe program is 
Hade Diagram
The gain and phase margins from the Bode plot are 
lOilf = U d B a t4>=1 0 rad/s~l 
\PM = 97.5° at a> = 0.085 r a ^
The MATLAB codetodrawtheBode magnitude plot of the do sed loop
system IS
» s y s l= f e e d b a c k ( s y s , 1 ) 
» m arg in ( s y s l )
» g r id
The Bode plot obtained onexecutionofthe program is 
Bmle liia g n u n
The low GM is c aused by the resonance b eing close to is stabihty.
From the Bode plot of the do sed-loop system, the frequency of the lightly 
damped mode is at =-10 rad/s .
And thisisborneoutbythestep response that shows a hghtly damped o scillation 
at 1.6 Hz or 10 rad/s .
(b)
The MATLAB codetodrawtheBodeplotofthe system with the 
bending mode damping increased from ̂ = 0.02 to 0.04 is
» f l = [ l / 0 . 0 1 1 ] ;
» f 2 = [ l 0 . 2 4 1]1 
» f 3 = [ l / 1 0 0 0 . 0 0 / 1 0 1 ] ;
» d e n = c o n v ( f l , c o n v ( f 2 , f 3 ) ) ;
Step 7 of 21
» sy s= tfr (num, dan) 
» m arg in (sys)
» g r id
Step 8 of 21 -a.
The Bode plot obtained onexecutionofthe program is
Rede Diagram
Gm - 7.13 iIB (a l ID rad /sec), Pm - 97A deg (a l D.DII5 rad/sec)
Step 9 of 21
The gain and phase margins fromtheBodeplotare 
|g J / = 7 1 3 d B a ta i= 1 0 r^ ^ 
lfM =97.5°at ai=0.085rad^
Step 10 of 21
We see that the (jM has increased considerably because theresonantpeslc 
is well below magnitude 1 ; thus the system will be much b etter behaved
Step 11 of 21
The MATLAB codetodrawtheBodeplotofthe system with the 
frequency of the bending modeincreasedfrom nv ( f l , conv ( f2 , f 3 ) ) ;
» s y s = t f (num, den)
» m arg in (sys)
» g r id
Step 12 of 21 A
The Bode plot obtained onexecutionofthe program is 
Kudc IN agram
Step 13 of 21
The gain and phase margins from the Bode plot are 
|gM =7.17dB ata»= 2 0 rad/s I 
\PM = 97.5° at a>= 0.085 l a ^
And again the GM is much improved and the resonant peak is 
signifrcantly reduced from magnitude 1 .
Step 14 of 21
@ii)
By picking up = 1 the MATLAB code to draw the Bode plot is 
» num =0.7S*[4 0. 4 1 ] ;
» f l = [ l / 0 . 0 1 1 ] ;
» f 2 = [ l 0 .2 4 1 ];
» f3 = [ l /1 0 0 0 .0 4 /1 0 1 ] ;
» f 4 = [ l 1 ] ;
» d e n = c o n v (f l , c o n v (f2 , c o n v (f3 , f 4 ) ) ) ;
» s y s = t f (num, den)
» m arg in (sys)
» g r id
Step 15 of 21 A
The Bode plot obtained on execution of the program is
B odcD iagiaB i
Gm - 34.8 d ll (a l 8.62 lad /H v) , P m - 92.7 deg (a l U.U847 nid/icc)
Step 16 of 21
The gain and phase margins from the Bode plot are 
|gM = 34.8 dB at a = 8.62 rad/s I 
\PM = 92.7° at a>= 0.0847 r a ^
which are healthy margins and the resonant peak is again, well below 
magnitudel.
Step 17 of 21
The MATLAB code to draw the Bode plot with notch Biter with the 
transfer function,
1
Uoo J
» n u m = 0 .7 S * [4 0 . 4 1 ] ;
» f l = [ l / 0 . 0 1 1] ;
» f 2 = [ l 0 . 2 4 1] ;
» f 4 = [ 1 /1 0 0 1] ;
» d e n = c o n v ( f l , c o n v ( f 2 , c o n v ( f 4 , f 4 ) ) ) ; 
» s y s = t f ( n u m ,den)
» m a r g i n ( s y s )
» g r i d
Step 18 of 21 A
The Bodeplotofthe system with the given notch Biter is shown below.
Bode D ia g ra in
G m - S U dB (a l I M ra d /s e c ) . I ’m - 97.4 deg (a l D.083 ladA ec)
Step 19 of 21
The gain and phase margins from the Bode plot are 
lGA/ = 55.2dBatai = 100 r ^
\PM = 97.4° at ai = 0 085 r a ^
which are healthiest margins of all the designs since the notch Biter has 
essentially canceled the bending mode resonant peak.
Step 20 of 21
W
Generally, the notch Biter is very sensitive to where to place the notch zeroes in 
order to reduce the hghtly damped resonant peak So if you want to use to notch 
Biter, you must have a good estimation ofthelocaBonofthe bending mode 
poles and the poles must remain at that location for all aircraft conditions,.
On the other hand, thelowpassBlteris relatively robust to where to place its 
break point Evaluation of the margins with the bending mode frequency lowered 
by lO^will show a drastic reduction in the margins for the notch Biter and very 
httle reduction for the low Biter.
Step 21 of 21
The table comparing the GM, PM, closed-loop bending mode dancing ratio, and 
resonant peak anq/litude is
Low pass Biter Notch Biter
GM 34.8
(ar = 8.62 rad/s)
55.2
(ai = lOOrad/s)
PM 92.7°
(ai = 0.0847 rad/s)
97.4°
(ai = 0.085 rad/s)
Qosed loop bending 
Mode Damping ratio
= 0 . 0 2 = 0.04
Resonant peak 0.08 0.068
The magnitude plot of the closed-loop system for Low-pass Biter is 
B ede liia g n u n
The magnitude plot of the closed-loop system for Notch Biter is 
B ode D iag ram
(d)
■While increasing the natural damping of the sj/stem would be the best solution, it 
might be difficult and e^ensive to carry out. Likewise, increasing the frequency 
is e^ensive and makes the structure heavier, not a good idea in an aircraft. Of the 
remaining two options, it is a better design to use a low pass Biter because of its 
reduced sensitivity to mismatches in the bending mode frequency.
Therefore the best recommendation would betousethelowpass Biter.