Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
1
Slid
es o
f Lec
ture
8
Toda
y’s
Cla
ss:
Rev
iew
Of H
ome
wor
k F
rom
Lec
ture
7H
amilt
on’s
Prin
cipl
eM
ore
Exa
mpl
es O
f Gen
eral
ized
Coo
rdin
ates
Cal
cula
ting
Gen
eral
ized
For
ces
Via
Vir
tual
Wor
k
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
2
Hom
ew
ork
from
Lec
ture
7
In th
e m
ost r
ecen
t cla
ss, w
e de
rived
the
gove
rnin
geq
uatio
ns f
or th
e co
mpo
und
pend
ulum
. The
hom
ew
ork
assi
gnm
ent
was
tove
rify
the
deriv
atio
nan
dto
linea
rize
the
resu
lting
equ
atio
ns.
We
foun
d th
e go
vern
ing
equa
tions
to b
e
and
R1
R2
m1
m2
θ 2
θ 1
m1
m2
+(
)R12θ 1
m2R
1R
2θ 2
θ 2θ 1
–(
)co
s+
m–2R
1R
2θ 22
θ 2θ 1
–(
)si
ng
m1
m2
+(
)R1
θ 1si
n+
0=
m2R
22θ 2
m2R
1R
2θ 1
θ 2θ 1
–(
)co
s+ m2R
1R
2θ 12
θ 2θ 1
–(
)si
ng
m2R
2θ 2
sin
+0
=+
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
3
Hom
ew
ork
from
Lec
ture
7
1.E
stab
lish
the
loca
tion
of s
tab
le e
quili
briu
m a
bout
whi
ch
to
linea
rize.
By
obse
rva
tion,
that
loca
tion
is.
2.S
ubst
itute
and
into
the
gove
rnin
g eq
uatio
ns a
nd e
xpan
d w
ith T
aylo
r se
ries.
and
θ 1θ 2
0=
=
θ 1θ 1s
∆θ 1
+=
0
θ 2θ 2s
∆θ 2
+=
0
m1
m2
+(
)R12θ 1
m2R
1R
2θ 2
θ 2θ 1
–(
)co
s+
m–2R
1R
2θ 22
θ 2θ 1
–(
)si
ng
m1
m2
+(
)R1
θ 1si
n+
0=
1
θ 1θ 2
-θ1
m2R
22θ 2
m2R
1R
2θ 1
θ 2θ 1
–(
)co
s+
m2R
1R
2θ 12
θ 2θ 1
–(
)si
ng
m2R
2θ 2
sin
+0
=+
1
θ 2θ 2
-θ1
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
4
Hom
ew
ork
from
Lec
ture
7
3.D
elet
e al
l ter
ms
invo
lvin
g po
wer
s an
d pr
oduc
ts o
f,
,, a
nd
and
θ 1θ 2
θ 1θ 2 m1
m2
+(
)R12θ 1
m2R
1R
2θ 2
+
m–2R
1R
2θ 22
θ 2θ 1
–(
)g
m1
m2
+(
)R1θ 1
+0
=0
m2R
22θ 2
m2R
1R
2θ 1
+
m2R
1R
2θ 12
θ 2θ 1
–(
)g
m2R
2θ 2
+0
=+
0
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
5
Hom
ew
ork
from
Lec
ture
7
4.G
roup
term
s to
for
m th
e m
ass
and
stiff
ness
mat
rix.
Not
e th
at b
oth
the
mas
s an
d st
iffne
ss m
atric
es a
re s
ymm
etric
and
posi
tive-
defin
ite.
The
mas
s eq
uatio
n w
ill b
e po
sitiv
e so
long
as
we
avo
id m
assl
ess
degr
ees
of fr
eedo
m, t
hat i
s w
e al
way
s w
ant t
o c
hoos
e ou
r de
gree
s of
free
dom
so
that
.
The
stiff
ness
mat
rixw
illal
way
sbe
non-
nega
tive
defin
ite.I
fth
ere
are
norig
id-b
ody
mod
es,
itw
illbe
posi
tive
defin
ite.W
esh
all
disc
uss
this
mor
ela
ter.
m1
m2
+(
)R12
m2R
1R
2
m2R
1R
2m
2R
22
θ 1 θ 2
gm
1m
2+
()R
10
0g
m2R
2
θ 1 θ 2+
0 0=
q r∂∂T
0≠
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
6
Ham
ilton
’s P
rinci
ple
We
cons
ider
a m
echa
nica
l sys
tem
who
se c
onfig
urat
ion
at a
ny
time
is
char
acte
rized
by
the
N g
ener
aliz
ed c
oord
inat
es. T
he s
yste
m is
subj
ect t
o po
tent
ial e
ner
gy a
nd a
ddi
tiona
l for
ces
and
evo
lves
over
the
inte
rva
l a
ccor
ding
to th
e La
gran
ge
equa
tions
for
eac
h.
We
can
ima
gine
the
evo
lutio
n of
the
syst
emco
nfigu
ratio
nov
erth
atin
terv
alby
pict
urin
gth
em
otio
nof
a po
int w
hose
coo
rdi
nate
s ar
e
in a
n N
-dim
ensi
onal
Car
tesi
an s
yste
m.
q{}
VF
rA{
}t 1
t 2,(
)
tdd
q r∂∂T
q r
∂∂T–
q r∂∂V
+F
rA=
r
q(t 2
)
q(t 1
)
q(t)
q1
q2
q3
qt()
{}
Ad
van
ced
Vib
rati
on
s
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e/dj
sega
l/UN
M/V
ibC
ours
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ides
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ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
7
Ham
ilton
’s P
rinci
ple
We
cont
ract
thes
esc
alar
equa
tions
with
test
func
tions
whi
ch
are
as
yetu
ndet
erm
ined
exce
ptfo
rth
eco
nditi
ons
for
each
r, a
nd th
en s
um th
em.
η rη r
t 1()
η rt 2(
)0
==
tdd
q r∂∂T
q r
∂∂T–
q r∂∂V
+F
rA–
η rt()
td
t 1t 2 ∫r
1=N ∑
0=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
8
Ham
ilton
’s P
rinci
ple
Inte
grat
ion
by
part
s yi
elds
Rec
allin
g th
at, t
he a
bove
inte
gral
sim
plifi
es
q r∂∂T
–q r
∂∂V+
FrA
–η r
t()
td
t 1t 2 ∫r
1=N ∑
tdd
q r∂∂T
ηr
q r∂∂T
η r–
td
t 1t 2 ∫r
1=N ∑
+0
=
η rt 1(
)η r
t 2()
0=
=
q r∂∂T
–q r
∂∂V+
FrA
–η r
t()
q r∂∂T –
η rî
td
t 1t 2 ∫r
1=N ∑
0=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
9
Ham
ilton
’s P
rinci
ple
The
abo
ve is
true
for
all
test
func
tions
.
Let
whe
re is
ano
ther
pat
h
from
to“n
ear”
the
path
take
n b
y.
Our
inte
gral
s ca
n no
w b
e w
ritte
n:
Obs
erve
that
the
term
s in
volv
ing
pote
ntia
l ene
rgy
are
a c
ompl
ete
diff
eren
tial.
η r
q(t 2
)
q(t 1
)
{q(t
)}~ {q(t
)}
q1
q2
q3
η rt()
q rt()
q rt()
–=
δqr
t()
=q
t()
{}
qt 1(
){
}q
t 2()
{}
qt()
{}
q r∂∂T
δqr
q r∂∂T
δqr
+
–
q r∂∂V
FrA
–δq
r+
î
td
t 1t 2 ∫r
1=N ∑
0=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
10
Ham
ilton
’s P
rinci
ple
Rea
rran
ging
the
abo
ve:
Whe
re.
Thi
s fo
rm o
f Ham
ilton
’s
prin
cipl
e as
ser
ts th
at th
e ac
tual
pat
h is
one
abou
t whi
ch
q r∂∂T
δqr
q r∂∂T
δqr
+
q r
∂∂V–
FrA
+δq
r+
î
r1
=N ∑
td
t 1t 2 ∫
δTδV
–δW
+(
)td
t 1t 2 ∫0
==
δWt()
FrA
t()
q rt()
q rt()
–(
)= δT
δV–
δW+
()
td
t 1t 2 ∫0
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
11
Ham
ilton
’s P
rinci
ple:
Spe
cial
Cas
e
For
the
spec
ial c
ase
whe
re th
e g
ener
aliz
ed f
orce
s a
re p
resc
ribed
load
s, w
e ca
n de
fine
the
“Pot
entia
l Ene
rgy
of L
oadi
ng”
and
the
“Tot
al P
oten
tial E
ner
gy”
is.
In th
is c
ase
, Ham
ilton
’s P
rinci
ple
beco
mes
: The
true
pat
h in
confi
gura
tion
spac
e of
the
syst
em m
akes
the
quan
tity
sta
tiona
ry.
FrA
AF
rAq r
r1
=N ∑–
=
VT
VA
+=
JT
VT
–(
)td
t 1t 2 ∫=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
12
Ham
ilton
’s P
rinci
ple:
Exa
mpl
eB
eam
Ben
ding
Equ
atio
n
The
str
ain
ener
gy in
an
Eul
er-
Ber
noul
li be
am is
. The
pote
ntia
l ene
rgy
of l
oadi
ng is
And
the
Kin
etic
Ene
rgy
is w
here
is th
e m
ass
per
unit
leng
th o
f the
bea
m.
M0 Q
0
ML
QL
p(x)
L
V1 2---
EI
x2
2 ∂∂y
2
xd
0L ∫=
AM
0x∂∂y
0M
Lx∂∂y
L–
Q0y
0()
–Q
Ly
L()
px()y
x()
xd
0L ∫–+
=
T1 2---
mt∂∂y
2
xd
0L ∫=
m
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
13
Ham
ilton
’s P
rinci
ple:
Exa
mpl
eB
eam
Ben
ding
Equ
atio
n
Lets
eva
luat
e th
e vi
rtu
al q
uant
ities
, beg
inni
ng w
ithK
inet
ic E
ner
gy:
and
δT1 2---
mt∂∂
yδy
+(
)2
x1 2---
mt∂∂y
2
xd
0L ∫–
d
oL ∫=
mt∂∂y
t∂∂δy
xd
0L ∫≅
δTtd
t 1t 2 ∫m
t∂∂y
t∂∂δy
xd
0L ∫
td
t 2t 2 ∫=
mt∂∂
t∂∂yδy
yδy
–td
t 1t 2 ∫
xd
0L ∫=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
14
Ham
ilton
’s P
rinci
ple:
Exa
mpl
eB
eam
Ben
ding
Equ
atio
n
Str
ain
Ene
rgy
:
δTtd
t 1t 2 ∫m
yδy
[]
t 1
t 2x
mt22 ∂∂y
δyxd
td
0L ∫t 1t 2 ∫
–d
0L ∫=
0
δV1 2---
EI
x22
∂∂y
δy+
()
2
x1 2---
EI
x2
2 ∂∂y
2
xd
0L ∫–
d
0L ∫=
EIy
''x22
∂∂δy
xd
0L ∫≅
EI
y''
x∂∂δy
0LE
Iy'
''δy
()
0L–
EIy
IVδy
xd
0L ∫+
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
15
Ham
ilton
’s P
rinci
ple:
Exa
mpl
eB
eam
Ben
ding
Equ
atio
n
The
str
ain
ener
gy te
rm b
ecom
es:
δVtd
t 1t 2 ∫= E
Iy'
'x∂∂δy
0LE
Iy'
''δy
()
0L–
EIy
IVδy
xd
0L ∫+
td
t 1t 2 ∫
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
16
Ham
ilton
’s P
rinci
ple:
Exa
mpl
eB
eam
Ben
ding
Equ
atio
n
In a
sim
ilar
man
ner
, we
find
the
cont
ribut
ion
from
Pot
entia
l Ene
rgy
of
Load
ing
δAtd
t 1t 2 ∫
M0
x∂∂δy
0M
Lx∂∂δy
L–
Q0δy
0()
–Q
Lδy
L()
+td
t 1t 2 ∫=
px()δ
yxd
0L ∫td
t 1t 2 ∫–
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
17
Ham
ilton
’s P
rinci
ple:
Exa
mpl
eB
eam
Ben
ding
Equ
atio
n
We
can
grou
p te
rms.
We
star
t with
the
term
s in
volv
ing
,,
, and
δy0(
)δy
L()
x∂∂δy
0x∂∂δy
L
M0
x∂∂δy
0M
Lx∂∂δy
L–
Q0δy
0()
–Q
Lδy
L()
+td
t 1t 2 ∫
EI
y''
x∂∂δy
0LE
Iy'
''δy
()
0L–
td
t 1t 2 ∫+
0=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
18
Ham
ilton
’s P
rinci
ple:
Exa
mpl
eB
eam
Ben
ding
Equ
atio
n
Fro
m w
hic
h w
e de
duce
.
A g
eom
etric
bou
ndar
y co
nditi
on s
peci
fyin
g im
plie
s th
at
. If d
ispl
acem
ent i
s no
t spe
cifie
d th
ere
, the
n
. Thi
s is
a “
natu
ral”
boun
dar
y co
nditi
on.
Sim
ilar
inte
rpre
tatio
ns a
re m
ade
of,
and
,
Q0
E–Iy
'''0(
)[
]δy
0()
0=
y0(
)δy
0()
0=
Q0
EIy
'''0(
)=
QL
E–Iy
'''L(
)[
]δy
L()
0=
M0
E–Iy
''0(
)[
]x∂∂δy
00
=M
LE–
Iy''
L()
[]
x∂∂δy
L0
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
19
Ham
ilton
’s P
rinci
ple:
Exa
mpl
eB
eam
Ben
ding
Equ
atio
n
Mat
chi
ng te
rms
in th
e sp
acia
l int
egra
l we
have
from
whi
ch
we
conc
lude
that
my
EIy
IVp
–+
[]δ
yx
t,(
)xd
0L ∫td
t 1t 2 ∫0
=
my
EIy
IV+
px
t,(
)=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
20
Ham
ilton
’s P
rinci
ple
Ham
ilton
’s p
rinci
ple
is g
ener
al a
nd a
lway
s w
orks
, tho
ugh
som
etim
es it
is h
ard
to e
valu
ate
.
In p
artic
ular
, not
e ho
w H
amilt
on’
s P
rinci
ple
is u
sed
to d
eriv
e th
e pa
rtia
ldi
ffer
entia
l go
vern
ing
equa
tions
.
Als
o, w
e sa
w h
ow
to d
efine
the
pote
ntia
l ene
rgy
of l
oadi
ng a
nd to
use
that
with
Ham
ilton
’spr
inci
ple
.We
will
see
that
we
can
also
use
itin
with
Lagr
ang
e’s
equa
tions
.
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
21
Cha
nge
of S
ubje
ct.
The
fol
low
ing
is a
n in
trod
uctio
n to
the
met
hod
ofA
SS
UM
ED
MO
DE
S.
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
22
Mor
e O
n G
ener
aliz
ed D
egre
es o
f Fre
edom
Dis
trib
uted
Dis
plac
emen
t
Lets
cons
ider
anE
uler
Ber
noul
libe
am s
impl
y su
ppor
ted
at e
ach
end.
Initi
ally
, we
assu
me
that
all
forc
es a
re c
onse
rva
tive
. We
post
ulat
e a
disp
lace
men
tdi
strib
utio
n of
the
sor
t
We
shal
l der
ive
Lagr
ang
e eq
uatio
ns f
or th
e e
volu
tion
of a
nd
. The
se a
re o
ur g
ener
aliz
ed d
egre
es o
f fre
edom
.L
yx
t,(
)A
1t()x
Lx
–(
)L
2----
--------
--------
A2
t()x2
Lx
–(
)L
3----
--------
--------
--+
=
A1
t()f
1x()
A2
t()f
2x()
+=
A1
t()
A2
t()
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
23
Mor
e O
n G
ener
aliz
ed D
egre
es o
f Fre
edom
Dis
trib
uted
Dis
plac
emen
t
Kin
etic
Ene
rgy
:
whe
re,
and
T1 2---
my2
xd
0L ∫1 2---
mA
1t()f
1x()
A2
t()f
2x()
+[
]2xd
0L ∫=
= 1 2---A
1(
)2I 1
2A
1A
2I 2
A2
()2
I 3+
+[
]=
I 1m
f 1x()2
xd
0L ∫m
L30--------
==
I 2m
f 1x()f
2x
xd
0L ∫m
L60--------
==
I 3m
f 2x()2
xd
0L ∫m
L10
5----
-----=
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
24
Mor
e O
n G
ener
aliz
ed D
egre
es o
f Fre
edom
Dis
trib
uted
Dis
plac
emen
t
Str
ain
Ene
rgy
:
whe
re
, a
nd
V1 2---
EI
y''
()2
xd
0L ∫1 2---
EI
A1
t()f
1''
x()
A2
t()f
2''
x()
+[
]2xd
0L ∫=
= 1 2---A
1(
)2I 4
2A
1A
2I 5
A2
()2
I 6+
+[
]=
I 4E
If 1
''x()2
xd
0L ∫E
I4 L3
--------
-=
=I 6
EI
f 2''
x()2
xd
0L ∫E
I4 L3
--------
-=
=
I 5E
If 1
''x()f
2''
x()
xd
0L ∫E
I2 L3
--------
-=
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
25
Mor
e O
n G
ener
aliz
ed D
egre
es o
f Fre
edom
Dis
trib
uted
Dis
plac
emen
t
Lagr
ang
e E
quat
ions
:
and
In m
atrix
for
m:
.
Not
e th
at b
oth
mat
rices
are
sym
met
ric, p
ositi
ve d
efini
te.
tdd
A1
∂∂T
A1
∂∂T–
A1
∂∂V+
A1I 1
A2I 2
A1I 4
A2I 5
++
+0
==
tdd
A2
∂∂T
A2
∂∂T–
A2
∂∂V+
A1I 2
A2I 3
A1I 5
A2I 6
++
+0
==
mL
15--------
1 2---1 4---
1 4---1 7---
A1
A2
EI
L3
------
42
24
A1
A2
+0 0
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
26
Gen
eral
ized
For
ces,
Cal
cula
ted
by
Met
hod
of V
irtu
al W
ork
Rec
all t
hat t
he g
ener
aliz
ed f
orce
ass
ocia
ted
with
the
gen
eral
ized
coor
dina
te is
We
exam
ine
the
incr
emen
tal w
ork
asso
ciat
ed w
ith in
crem
ents
of
:
The
gen
eral
ized
for
ce a
ssoc
iate
d w
ith th
e g
ener
aliz
ed c
oord
inat
e is
q rF
rF
nq r
∂∂xn
⋅n∑
=
q r
δWF
rδq r
Fn
q r∂∂x
nδq
r⋅
n∑F
nδx
n⋅
n∑=
==
q r
Fr
δW δqr
--------
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
27
Gen
eral
ized
For
ces,
Cal
cula
ted
by
Met
hod
of V
irtu
al W
ork
Lets
cons
ider
anE
uler
Ber
noul
libe
am s
impl
y su
ppor
ted
at e
ach
end.
We
cons
ider
mom
ents
M0
and
ML a
pplie
d at
the
ends
and
adi
strib
uted
trac
tion
appl
ied
alon
g th
e le
ngth
of t
he b
eam
.
We
post
ulat
e a
disp
lace
men
t dis
trib
utio
n of
the
sor
t
Lets
cal
cula
te th
e g
ener
aliz
ed f
orce
s as
soci
ated
with
the
gen
eral
ized
coor
dina
tes
and
.
M0
ML
p(x)
L
yx
t,(
)A
1t()x
Lx
–(
)L
2----
--------
--------
A2
t()x2
Lx
–(
)L
3----
--------
--------
--+
=
A1
t()f
1x()
A2
t()f
2x()
+=
A1
A2
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
28
Gen
eral
ized
For
ces,
Cal
cula
ted
by
Met
hod
of V
irtu
al W
ork
The
wor
k do
ne to
the
stru
ctur
e b
y th
e fo
rces
act
ing
thro
ugh
a vi
rtu
al d
ispl
acem
ent
, is
so
can
be
calc
ulat
ed s
imila
rly.
δyδA
1(
)f1
x()
=
δWδA
1M
0f 1
'0(
)–
ML
f 1'
L()
px()f
1x()
xd
0L ∫+
+
= FA
1M
0f 1
'0(
)–
ML
f 1'
L()
px()f
1x()
xd
0L ∫+
+
=
FA
2
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
29
Hom
ew
ork
for
Lect
ure
8A
num
eric
al e
xper
imen
t with
line
ariz
atio
n
Man
y tim
es w
e ha
ve d
eriv
ed th
e eq
uatio
ns f
or a
spr
ing
rein
forc
ed p
endu
lum
:.
The
line
ariz
ed f
orm
is w
here
.
We
use
the
linea
rized
freq
uenc
y to
non
-dim
ensi
onal
ize
the
time
para
met
er. D
efine
, defi
ne, a
nd d
efinem
R
θ
κθ
κm
R2
--------
--θ
g R---θ
sin
+
+
0=
θω
L2θ
+0
=
ωL2
κm
R2
--------
--g R---
+=
τω
Lt
=φ
τ()
θτ
ωL
⁄(
)=
αg
R⁄ ωL2
--------
--=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
ours
e/sl
ides
/Lec
ture
8.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
30
Hom
ew
ork
for
Lect
ure
8co
ntin
ued
The
n a
nd
1.S
olve
num
eric
all
y th
e di
men
sion
less
go
vern
ing
equa
tion
for
the
initi
al c
ondi
tions
: a
nd o
ver
the
perio
d
for
the
thre
e ca
ses:
,, a
nd.
2.D
oth
esa
me
asab
ove
but
for
the
initi
alco
nditi
ons
and
3.C
ompa
re a
nd d
iscu
ss th
e y
our
resu
lts f
or p
arts
1 a
nd 2
.
τ22
ddφ
t22
ddθ
1 ωL2
------
=τ22
ddφ
1α
–(
)φα
φsi
n+
[]
+0
=
φ0(
)π
=τ
ddφ0
0=
06
π,
()
α0
=α
12⁄
=α
1=
φ0(
)π 6---
=
τddφ
00
=