Homework 2 Solution
Homework 2 Solution
Homework 2 Solution
O"
#ro$lem 1 (35 pts) Consider the very simple glider shown
at the right. It consists of a wing, tail and body. Each item
is rectangular, and each has the same thickness, t. Denote
the length and width of the wing as
w
l
and
w
w
, respectively.
Use similar notation for the dimensions of the tail and the body.
(a)(15 pts) Develop the expression for the location of the cg , call it
cg
x
, along the x-axis.
Solution
Using the diagram at the right, since the cg is that location
about which the net moment is !ero, we have
" # $ % & # $ % & # $ % & + +
t cg t cg b w cg w
w x W x W w x W
or
# $ % & # $ % & # $ % &
t t b w w cg tot
w W W w W x W + +
or
tot
t t b w w
cg
W
w W W w W
x
# $ % & # $ % & # $ % & + +
% &'#
($)(10pts) (ssume that xcg is given. )urthermore, assume that the wing and tail widths are fixed, and that
the wing length is also fixed. )ind the expression for the tail length as a function of these elements, so that
the glider will have positive stiffness for small changes in the angle of attack, . *ou should assume that
the speed is sufficiently slow that the backwash onto the tail region is negligible. (lso, assume that drag
moments are negligible.
Solution In addition to the stated assumptions, we will make the following assumptions
&('# +he body contributes !ero lift.
&($# +he wing lift force is # &
$
V S L
w w
and the tail lift is # &
$
V S L
t t
.
+o compute the angle of attack,
,
"
, then
the airplane is said to have negative pitc# stiffness.
(a)(5pts) Discuss how Definition $.$ compares to the definition given in the discussion at the website
http%%books.google.com%booksF
idG+"Ux=avHbtIC;pgG?(=1;lpgG?(=1;d/GdefinitionIpositiveIpitchIstiffness;sourceGbl;otsGku=
gx9J$mf;sigG4KHiJBsA6cLMN6??nphOf!I9fP";hlGen;eiG*Jda+pbbDumBsQA)=pysD(;saGN;
oiGbookPresult;ctGresult;resnumG6;s/iG$;vedG"CDEQ=(Ew8(RvGonepage;/;fGfalse
Solution +o /uote the site
D"#e condition of a steady state moment tending to !esto!e euilib!ium is $nown as positive pitch
stiffness, and, in t#is case,positive pitc# stiffness is associated wit# a negative slope of t#e pitc#ing
moment ve!sus alp#a cu!ve.E
+his definition is essentially the same as that given in Definition $.$. +he case referred to is a special case
of the more general definition.
($)(5pts) Discuss Definition $.$ in relation to Figure 2.4.
Solution+ In )igure $.H neither curve represents an aircraft having an e/uilibrium position &i.e. they are
unbalanced#. 5ince Definition $.$ is predicated on the assumption that an e/uilibrium state exists, )igure
$.H is not relevant, per this definition.
(&)(5pts) Bn p.$' of the text the authors state &essentially# D%f t#e pitc# stiffness is to be positive at t#e
euilib!ium angle of attac$,
o
, t#en &i#
# &
o m
C
must be &e!o, and &ii#
# &
m
C
must be negative.
Explain why &i# is not a needed condition in order for pitch stiffness to be positive &in the sense of
Definition $.$ above#.
Solution ?er Definition $.$, all that is needed as that the moment be restoring. .athematically, this
definition re/uires that
"
" # & < > fo! C
m
and
"
" # & > < fo! C
m
. +hese conditions re/uire no
conditions on
% # &
m m
C C
a
, other than, &by definition#
" S % # &
"
<
m m
C C
a
&assuming a
uni/ue value for
"
.#
(,)(5pts) Explain why condition &ii# in &c# is not a necessary condition for an airplane to have positive
pitch stiffness &in the sense of Definition $.$ above#.
Solution +he explanation is given in part &c#. Condition &ii# is a sufficient condition, but not necessa!y.
(e)(5pts) Consider an airplane with
# &
m
C
as shown at the right.
Clearly, this airplane has
" # &
o m
C
and
"
# &
<
o
m
C
.
7owever, unlike the curve &a# in 'igu!e (.) on p.$" of the text,
we do not have "
# &
<
m
C
for all . In the sense of Definition
$.$ above, explain why the airplane has positive stiffness.
*xplanation It has positive pitch stiffness per Definition $.$ because it satisfies the ine/ualities given in
part &c#.
E-tra .re,'t (5pts) In view of the example in &e#, determine the mathematical condition on
# &
m
C
for
an airplane to have positive pitch stiffness &per Definition $.$ above#.
Solution+ +he mathematical conditions were given by the ine/ualities in part &c#.
E-tra .re,'t (5pts) Identify an airplane that has positive stiffness, but does not have "
# &
<
m
C
for
all . Include reference&s# to support your identified airplane.
Solution I have no idea
o
# &
m
C
#ro$lem 3 (15 pts) Carry out the mathematical steps described by the authors on p.$L of the text to arrive
at e/uation &$.1.6#. In doing so, be careful to show more details that are given in the text.
Solution
.
# & # &
# &
p t
wb
wb
ac
m L
, n L
m
m
C C
V # # C
C
C &$.1.$#
If the true aerodynamic center of the wing%body exists, the term
"
wb
ac
m
C
. Even if it does not exist in
the classical sense, the neutral location,
n
#
for the entire airplane does exist. 7ence, from Definition $.H,
for
n
# #
, &$.1.$# becomes
.
# & # &
# & "
p
t
wb
wb
ac
m L
, n n L
m
C C
V # # C
C
solving this e/uation for
n
#
gives
.
# &
# &
'
1
1
]
1
p
t wb
ac
wb
m
L
,
m
L
n n
C
C
V
C
C
# # &$.1.H#
Even though &$.1.H# gives the expression for
n
#
, this re/uires knowledge of three pitch derivatives, two
of which vary with . (s an alternative to this approach to determining
n
#
, we can re2write &$.1.H# as0
.
# &
# &
'
1
1
]
1
p
t wb
ac
wb
m
L
,
m
L
n n
C
C
V
C
C
# # &$.1.HT#
5ubstituting &$.1.HT# back into &$.1.$# gives
# &
n L m
# # C C
. &$.1.6#
#ro$lem / (10 pts) Oo to the website
http%%books.google.com%booksF
idG+"Ux=avHbtIC;pgG?(=1;lpgG?(=1;d/GdefinitionIpositiveIpitchIstiffness;sourceGbl;otsGku=
gx9J$mf;sigG4KHiJBsA6cLMN6??nphOf!I9fP";hlGen;eiG*Jda+pbbDumBsQA)=pysD(;saGN;
oiGbookPresult;ctGresult;resnumG6;s/iG$;vedG"CDEQ=(Ew8(RvGonepage;/;fGfalse
and from the material there, develop a definition of the ae!odynamic cente! .
-nswe! De)'*'t'o*0 +he aerodynamic center &ac# is that point along the x-axis about which the pitch
moment tends to be invariant with respect to . -:ote It is normally close to the vehicle mean chord line0
about U of the distance behind the leading edge for subsonic speed, and then moving toward the V
distance for supersonic speed.,
#ro$lem 5 (15 pts) +ext ?roblem $.' parts &a#, &b#, and &c#. ?lease restate the problem and each part
7E@E.
( subsonic transport plane has a tapered, untwisted, sweptback wing with straight leading and trailing
edges. +he wing tips are straight and parallel to the root chord. In the following, use (ppendix C, and
assume that the airfoil section local aerodynamic center &ac# is at the U2chord point.
(a)(5pts) .ake an accurate 12view drawing of the wing chord plane.
($)(5pts) Calculate &i# wing area, &ii# aspect ratio, &iii# taper ratio,
! t
c c %
, and &iv# mean aerodynamic
chord, c .
Solution
&i# wing area2 +he area of a trape!oid is b
c c
-
t !
,
_
$
. (nd so
$
$446 46
$
'$ $6
$ ft S
,
_
+
.
&ii# aspect ratio )rom p.1=", '' . 3 $446 % '6" %
$ $
S b - .
&iii# taper ratio
H3 . " $6 % '$ %
! t
c c
.
&iv# mean aerodynamic chord
ft
c
c
!
$= . 'L
'
'
1
$
$
,
_
+
+ +
W '6"
W '$
W $6
o
$=
"./
S%0*
1."".M
W 6
W LL . $"
W 1$ . $
(&)(5pts) Calculate the location of the wingTs mean aerodynamic center. +hen locate it and c on the side
view of the wing diagram in &a# &including dimensions#. -(ssume a uniform lift coefficient,
L l
C C
,
Solution
ft
b
y '' . 11
# ' & 1
$ '
$
,
_
+
+
-7alf2wing position,
H % '
tan
H
+ y
c
x
!
where
o
- - L= . $1
'
'
# tan& # tan&
" H % '
+
. 7ence,
ft x LL . $"
$ %
"
$ %
"
$ $
b b
l
L
c&dy
S
c&dy C
S C
&
a
where
y y y &
y y y c c
o
"==L . # H tan& # &
'41 . $6
46
'1
$6 # &
,
_
7ence,
ft y y & 1$ . $
1
"'"$L .
$
4H3 . '
$446
$
46
"
1 $
1
]
1