16.060 Principles of Automatic Control
Final Exam
Fall 2002
Part A: Put your working and answers in the booklet provided.
Question A.1
20 points
Consider the plant
G(s ) =
1.25
2
s + s + 1.25
=
1.25
(s + 0.5 j )(s + 0.5 + j )
a) The desired closed-loop pe

16.100 Take-Home Exam #1
Distributed: October 17 at 10am
Due: October 24 at 9am
Directions: This take-home exam is to be completed without help from any other individual
except Prof. Darmofal. This includes help from other students (whether or not they ar

Derivation of Sound Wave Properties
Sound wave
propagating to
right
+ d
du
p + dp
a
u = 0 ( x) =
p
Similar for u ( x), p( x)
xs
l
Assume:
Sound wave creates small disturbances in an isentropic manner.
Mass
dl
( x)dx + u l u 0 = 0
dt 0
l
d xs
( + d )dx

Behavior of Isentropic Flow in Quasi-1D
Recall cons. of mass:
uA = const.
Consider a perturbation in the area
R = L + d
L =
p R = p L + dp
pL = p
u R = u L + du
uL = u
AR = AL + dA
AL = A
xL
xR
uA = ( + d )(u + du )( A + dA)
= uA + duA + Adu + udA + H .

Critical Mach Number
We can estimate the freestream Mach number at which the flow first accelerates
above M > 1 (locally) using the Prandtl-Glauert scaling and isentropic
relationships.
Recall from P-G:
On the airfoil
surface:
C p (M ) =
C p ( M = 0)
2
1

Solution
2
y
V = 100 mph
8
3
=10 miles
1
u n = 0 at control pt #1:
The velocity at control pt #1 is the sum of the freestream + 3 point vortices
velocities at that point:
u1 = Vi +
1
2
3
i
i+
j
2
2
2
2
2
2
The normal at control pt #1 is:
n1 = i
u1 n1

Subsonic Small Disturbance Potential Flow v vv 1. V = (V + u )i + vj where | u2 + v2 | < 1 2 V small disturbances are assumed
vv 2. ui + vj = perturbation potential u = v= x y
3. small-disturbance (?) and bcs= 2 2 2 (1 M ) 2 + 2 = 0 y x
BC: v( x,0) = V d

Drag Tare Due to Mount
Force balance will measure drag which is due to exposed portion of mount:
extra drag
due to
exposed
mount
Forces
measured
here
fairing
mechanism
to adjust
Two techniques to estimate drag tare:
Remove model and run tests to find dr

Three-Dimensional Wall Effects
In a freestream, recall that a lifting body can e modeled by a horseshoe vortex:
V
Consider a rectangular cross-section tunnel:
Flow is into page
wing
The image system for this looks like:
images
images
images
images
actual

Waves in 1-D Compressible Flow
Imagine we have a steady 1-D compressible flow. Then suppose a small
disturbance occurs at a location x = xo . This disturbance will cause waves to
propagate away from the source. Suppose that the flow velocity were u and th

Normal Shock Waves
In our quasi-1D flows, shocks can occur from a supersonic-to-subsonic state.
These shocks are discontinuous in our inviscid flow model (recall that shocks are
very thin and their thickness scales with 1 ):
Re
M
ML
shock
1
Upstream Mach:

Propagation of Disturbances By a Moving Object
Consider an object moving at speed V :
Vt1
t3
t2
V t =0
t1
Suppose that the atmospheric speed of sound is a . The body emits sound
waves as it travels through the atmosphere and these wave propagate away
from

Linearized Compressible Potential Flow Governing Equation
Recall the 2-D full potential eqn is:
1
1
2
1 2 ( x ) 2 xx + 1 2 ( y ) 2 yy 2 x y xy = 0
a
a
a
1
2
Where a 2 = a o
( x ) 2 + ( y ) 2
[
]
As you saw, for small perturbations to a uniform flow, the

Implications of Linearized Supersonic Flow on Airfoil Lift & Drag
To begin, we will divide the airfoil geometry into camber and thickness
distributions:
y
yc ( x )
( x)
yu ( x)
x
yl ( x )
1
i
v
v
v
U U = u cos i + U sin j
1
y u ( x) = y c ( x) + ( x)
2
1

Oblique Shock Waves
Heres a quick refresher on oblique shock waves. We start with the oblique shock
as shown below:
w2 , M t 2
(1)
(2)
( )1 : upstream flow
u2 , M n 2
y
condition
u1 , M n1
w1 , M t1
( )2 : downstream flow
v2 , M 2
condition
x
: angle of

Prandtl-Meyer Expansion Waves When a supersonic flow is turned around a corner, an expansion fan occurs producing a higher speed, lower pressure, etc. in an isentropic process. fan Forward Mach line
M1 > 1
1 2
Rearward Mach line waves
Just as we saw with

Computational Methods for the Euler Equations
Before discussing the Euler Equations and computational methods for them, lets
look at what weve learned so far:
Method
2-D panel
Assumptions/Flow type
2-D, Incompressible, Irrotational Inviscid
Vortex lattice

Structured vs. Unstructured Grids The choice of whether to use a structured or an unstructured mesh is very problem specific (as well as company/lab specific). The answer is one of engineering judgement. Here are some of the issues: (1) Complex geometry:

Ground Effect Using Single Vortex Model
b
h
What is the boundary condition at ground ( z = 0 ) and does a single horseshoe
vortex satisfy it?
B.C.: solid wall
vv
u n = 0
w = 0 at z = 0!
Consider from far downstream:
z
h
b
So, to satisfy bc:
h
b
h
b
Image

Single Horseshoe Vortex Wing Model
S
b b
trailing vortices
bound vortex
Lift due to a horseshoe vortex Kutta-Joukowsky Theorem
b 2
L = V dy = V b
b 2
1 V2 S 2 2 b 2 CL = V b S
CL = 2 A V b
CL =
L
=
V b
1 V2 S 2
Single Horseshoe Vortex Wing Model
Induc

z
Thin Airfoil Theory Summary
(x) = thickness
z(x) = camber line
x
c
Replace airfoil with camber line (assume small
c
)
z
z(x) = camber line
x
c
Distribute vortices of strength ( x) along chord line for 0 x c .
z
(x)dx
x
c
Determine ( x) by satisfying flo

Important Concepts in Thin Airfoil Theory
1. This airfoil theory can be viewed as a panel method with vortex solutions
taking the limits of infinite number of panels & zero thickness & zero camber
cfw_ lim vortex panel = thin airfoil theory
lim
thickness

Force Calculations for Lifting Line
Recall:
N
( y ) = ( ) = 2bV An sin n
n =1
b
y = cos
2
The local two-dimensional lift distribution is given by Kutta-Joukowsky:
L ( y ) = V ( y )
N
L ( ) = 2bV2 An sin n
n =1
To calculate the total wing lift, we integr

Problem #1
Assume:
Incompressible
1
2-D flow Vz = 0, = 0
z
Steady
=0
t
Parallel Vr = 0
r1
r0
a) Conservation of mass for a 2-D flow is:
1
1
( r Vr ) +
(V ) = 0
r r
r
=0
(V ) = 0 V does not depend on
V = V ( r )
b) -mometum equation is:
V
VV
1 p
2 Vr

Viscous Flow: Stress Strain Relationship Objective: Discuss assumptions which lead to the stress-strain relationship for a Newtonian, linear viscous fluid:
ij =
ui u j u + + k x j xi ij xk uk u v w = + + = iV xk x y z
where = dynamic viscosity coeffici

Integral Boundary Layer Equations
Displacement Thickness
The displacement thickness * is defined as:
* = 1
0
u
u
dy = 1 dy
e ue
ue
0
compressible
flow
incompressible
flow
The displacement thickness has at least two useful interpretations:
Interpret

Correlation Methods for Integral Boundary Layers
We will look at one particularly well-known and easy method due to Thwaites in
1949.
First, start by slightly re-writing the integral b.l. equation. We had:
w
d
du e
=
+ (2 + H )
2
u e dx
e u e dx
Multipl

Method of Assumed Profiles
Here are the basic steps:
1. Assume some basic boundary velocity profile for u ( x, y ) . For example, this is a
crude approach but illustrates the ideas:
y
, 0 y < ( x)
u( x, y )
= ( x)
ue ( x )
1, y ( x )
where ( x) is the s

Falkner-Skan Flows
For the family of flows, we assume that the edge velocity, u e ( x) is of the
following form:
u e ( x) = Kx m
K = arbitrary constant
The pressure can be calculated from the Bernoulli in the outer, inviscid flow:
12
u e = const.
2
dp
du

Effect of Turbulent Fluctuations on Mean Flow: Reynolds-Averaging
In a turbulent flow, we can define the mean, steady flow as:
T
1
u ( x, y , z ) = lim u ( x, y , z , t )dt
T T
0
This allows us to split the flow properties into a mean and a fluctuating pa