home/djsegal/unm/vibcour - mech.utah.edubrannon/public/djsegal/vibclass/lecture10.frm... · ad v...
TRANSCRIPT
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
1
Slid
es o
f Lec
ture
10
Toda
y’s
Cla
ss:
Rev
iew
of H
ome
wor
k fr
om L
ectu
re 8
A s
hor
t qui
z on
line
ariz
atio
n.Li
near
izat
ion
of L
agr
ang
e E
quat
ions
Pro
pert
ies
of R
esul
ting
Mat
rix E
quat
ions
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
2
Hom
ew
ork
from
Lec
ture
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
dim
ensi
onle
ss th
e tim
e pa
ram
eter
.
Defi
ne, d
efine
, and
defi
ne
m
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
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
3
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
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
4
Sol
utio
nU
sing
Mat
lab
Usi
ng s
tate
spa
ce f
orm
ulat
ion:
, the
n
Thi
s ap
pear
s as
a fu
nctio
n ‘p
endu
1.m
’ pr
ovid
ed to
mat
lab:
func
tion
yprim
e =
pend
u1(t
,x)
glob
al a
lpha
;yp
rime
= [ -
alph
a.*s
in(x
(2))
- (
1-al
pha)
.*x(
2);
x(1)
];re
turn
;
xφ φ
=x
αx 2(
)1
α–
()x
2–
sin
–
x 1=
Advanced Vibrations
/home/djsegal/UNM/VibCour se/slides/11/30/98 Copyright Dan Segalman, 1998
5
ResultsCalls to Matlab
T = [0:299]*(6*pi)/200; %define time arra y% phi_max = pi; % Initial Condition φ=π% alpha = 1; [t,x] = ode23(’pendu1’, T , [0 phi_max]); %Rung e-Kuta Integration x1 = x;% alpha = 0.5; [t,x] = ode23(’pendu1’, T , [0 phi_max]); x2 = x;% alpha = 0.0; [t,x] = ode23(’pendu1’, T , [0 phi_max]); x3 = x;% plot(t,x1(:,2), t,x2(:,2), t,x3(:,2)); print pi -depsc%% phi_max = pi/6;% alpha = 1; [t,x] = ode23(’pendu1’, T , [0 phi_max]); x1 = x;% alpha = 0.5; [t,x] = ode23(’pendu1’, T , [0 phi_max]); x2 = x;% alpha = 0.0; [t,x] = ode23(’pendu1’, T , [0 phi_max]); x3 = x;% plot(t,x1(:,2), t,x2(:,2), t,x3(:,2)); print pi6 -depsc
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
6
Sol
utio
n f
orφ 0
=π
Whe
n th
e pe
ndul
um s
tar
tsdi
spla
ced
near
ly
vert
ical
ly,
the
rest
orin
g f
orce
due
togr
avi
ty is
als
o ne
ar z
ero.
The
mor
e of
the
rest
orin
g f
orce
that
is d
ue to
the
tor
sion
al s
trin
g, th
em
ore
that
the
resp
onse
will
app
ear
to b
e ha
rmon
ic.
α=0.5
α=0.
00
510
1520
2530
−4
−3
−2
−101234
α=1
κ θ
R
m
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
7
Sol
utio
n f
orφ 0
=π/6
Whe
n th
epe
ndul
um s
tar
tson
ly a
littl
edi
stan
ce fr
om th
est
atic
equ
ilibr
ium
,th
e ‘s
mal
l ang
le’
appr
oxim
atio
n is
ver
y go
od.
In th
at c
ase
, the
line
arity
of
the
resu
lt is
nea
rly
inde
pend
ent o
f whe
ther
the
rest
orin
g f
orce
is d
ue to
the
sprin
g or
to g
ravi
ty.
Legi
timat
e lin
eariz
atio
n re
quire
s th
at th
e lin
eariz
ed te
rms
be n
earl
yeq
ual t
o th
e co
rres
pond
ing
nonl
inea
r te
rms
thr
ough
out t
hede
form
atio
n.
α=0.5
α=0.
0α=
10
510
1520
2530
−0.
8
−0.
6
−0.
4
−0.
20
0.2
0.4
0.6
m
R
θ
κ
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
8
Sho
rt Q
uiz
Thi
s qu
iz s
houl
d ta
ke a
bout
15
min
utes
.
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
9
For
mal
Lin
eariz
atio
n of
La
gran
ge
Equ
atio
nsD
eriv
atio
n of
Sys
tem
Mat
rices
Inth
elim
iting
case
sof
smal
lde
flect
ion
(),
smal
lra
tes
,
the
Lagr
ang
e eq
uatio
ns s
impl
ify.
We
begi
n b
y lo
okin
g at
kin
etic
ene
rgy
.
Aga
in,
we
mak
eus
eof
one
ofth
eco
reob
serv
atio
nsin
the
deriv
atio
nof
the
Lagr
ang
e eq
uatio
ns:
q r1
«q r
1«
Tm
n 2------
x˙ nx˙ n
⋅n∑
mn 2------
q r∂∂x
˙ n
q s∂∂x
˙ n⋅
q rq
ss∑
r∑n∑
==
q r∂∂x
˙ n
q r∂∂x
n=
Tm
n 2------
q r∂∂x
n
q s∂∂x
n⋅
q
rqs
s∑r∑
n∑1 2---
q rqsM
rsq{
}(
)s∑
r∑=
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
10
For
mal
Lin
eariz
atio
n of
La
gran
ge
Equ
atio
nsD
eriv
atio
n of
Sys
tem
Mat
rices
In th
e lim
it of
sm
all d
ispl
acem
ent a
bout
the
confi
gura
tion
of s
tab
le
equi
libriu
m, t
he m
ass
mat
rix b
ecom
es
.
Not
e th
at is
stil
l sym
met
ric.
Impo
rtan
t: In
the
limit
of s
mal
l dis
plac
emen
t,
&
qs{
}
Mrs
Mrs
qs{
}(
)m
n 2------
q r∂∂x
n
qs
{}
q s∂∂x
n
qs
{}
⋅
n∑=
=
Mrs
Mrs
qs{
}(
)=
T1 2---
q rq sMrs
s∑r∑
=M
rsq r
q s∂
2 ∂∂T
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
11
For
mal
Lin
eariz
atio
n of
La
gran
ge
Equ
atio
nsD
eriv
atio
n of
Sys
tem
Mat
rices
Obs
erve
that
the
mas
s m
atrix
is p
ositi
ve s
emi-d
efini
te. C
onsi
der
an
arra
yof
gen
eral
ized
spee
ds.T
heco
rres
pond
ing
kine
ticen
ergy
isqt
{}
Tqt
{}
()
1 2---M
rsq rt q st
rs,∑
=
1 2---q rt q st
mn 2------
q r∂∂x
n
qs
{}
q s∂∂x
n
qs
{}
⋅
n∑r
s,∑=
mn 2------
q r∂∂x
n
qs
{}
q s∂∂x
n
qs
{}
⋅
q rt q st
s∑r∑
n∑=
mn 2------
y˙ nty˙ nt
0w
here
y˙ nt≥
⋅n∑
q r∂∂x
n
qs
{}q rt
r∑=
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
12
For
mal
Lin
eariz
atio
n of
La
gran
ge
Equ
atio
nsD
eriv
atio
n of
Sys
tem
Mat
rices
The
mea
ning
of t
he m
ass
mat
rix.
Con
side
r th
e “a
ccel
erat
ion”
for
ce
.
Nex
t, co
nsid
er c
olum
n ve
ctor
of g
ener
aliz
ed a
ccel
erat
ions
that
are
all
zero
sex
cept
for
a“1
”on
the
r’th
row
:.T
he
resu
lting
“ac
cele
ratio
n f
orce
s” s
een
by
eac
h of
the
othe
r g
ener
aliz
ed
degr
ees
of fr
eedo
m a
re th
e r’t
h co
lum
n of
.
tdd
q r∂∂T
t
ddq r
∂∂1 2---
Mst
q sqt
t∑s∑
Mrs
q ss∑
==
q{} r
0.
1.
.0
T=
M
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
13
For
mal
Lin
eariz
atio
n of
La
gran
ge
Equ
atio
nsE
xam
ple
of M
ass
Mat
rix
Lets
con
side
r a
stru
ctur
esi
mila
rto
the
disk
-sha
ftsy
stem
inM
eiro
vitc
h.
Igno
ring
the
kine
tic e
ner
gy o
fth
e sh
aft,
the
kine
tic e
ner
gy o
fth
e sy
stem
s is
.
Our
mas
s m
atrix
is
L1
L2
L4
L3
J 1,θ
1J 2
,θ2
J 3,θ
3
E, J
s
T1 2---
J 1θ 12
1 2---J 2
θ 221 2---
J 3θ 32
++
=
Mij
[]
θ iθ
j∂
2 ∂∂T
J 10
0
0J 2
0
00
J 3
==
Advanced Vibrations
/home/djsegal/UNM/VibCour se/slides/11/30/98 Copyright Dan Segalman, 1998
14
AnotherExample of Mass Matrix
Consider acantile vered beam f orwhic h we postulate adisplacement fieldwith tw o freegeneraliz edcoor dinates:
The kinetic ener gy in the beam will be
where ,
and
EI, m
L
y x t,( ) A1 t( ) f 1 x( ) A2 t( ) f 2 x( )+=
T A1 A2,( ) m2---- y( )2
xd
0
L
∫=
12--- A1 t( )
2I1 2A1 t( ) A2 t( )I2 2A2 t( )( )I3++[ ]=
I1 m f 1 x( )( )2xd
0
L
∫= I3 m f 2 x( )( )2xd
0
L
∫=
I2 m f 1 x( ) f 2 x( ) xd
0
L
∫=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
15
Ano
ther
Exa
mpl
e of
Mas
s M
atrix
Con
tinue
d
Our
mas
s m
atrix
isM
ij[
]A
iA
j∂
2 ∂∂T
mI 1
I 2I 2
I 3=
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
16
For
mal
Lin
eariz
atio
n of
La
gran
ge
Equ
atio
nsD
eriv
atio
n of
Sys
tem
Mat
rices
Con
side
rth
epo
tent
ial
ener
gyin
the
vici
nity
ofa
confi
gura
tion
ofst
able
equi
libriu
m. I
n an
equ
ilibr
ium
con
figur
atio
n, th
e su
m o
f the
pot
entia
l
forc
es in
eac
h di
rect
ion
is z
ero
.
Tayl
or s
erie
s e
xpan
sion
for
is
.
We
defin
e th
e st
iffne
ss m
atrix
to b
e.
Not
e th
at, b
y co
nstr
uctio
n, is
sym
met
ric.
0q r
∂∂V –=
V
Vq{
}(
)V
01 2---
q rq s
∂
2 ∂∂V
qs{
}∆q r∆
q sH
.O.T
.+
s∑r∑
+=
Krs
q rq s
∂
2 ∂∂V
qs{
}
=
K
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
17
For
mal
Lin
eariz
atio
n of
La
gran
ge
Equ
atio
nsD
eriv
atio
n of
Sys
tem
Mat
rices
Pot
entia
l ene
rgy
in te
rms
of S
tiffn
ess.
In th
e lim
it of
sm
all s
trai
ns,
whe
re w
e ha
ve r
edefi
ned
the
gen
eral
ized
coo
rdin
ates
to b
e z
ero
at th
e
equi
libriu
m c
onfig
urat
ion.
Usu
all
y, w
e se
t the
dat
um s
o th
at.
Bec
ause
sta
bilit
y re
quire
s th
at p
oten
tial e
ner
gy b
e at
a lo
cal m
inim
um,
for
all
. Thi
s is
an
asse
rtio
n th
at th
e st
iffne
ss
mat
rix is
pos
itive
sem
i-defi
nite
.
VV
01 2---
Krs
q rqs
s∑r∑
+=
V0
0=
Krs
q rqs
0≥
s∑r∑
q{}
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
18
For
mal
Lin
eariz
atio
n of
La
gran
ge
Equ
atio
nsD
eriv
atio
n of
Sys
tem
Mat
rices
The
mea
ning
of t
he S
tiffn
ess
mat
rix.
Con
side
r th
e g
ener
aliz
ed f
orce
.
Nex
t, co
nsid
er c
olum
n ve
ctor
of g
ener
aliz
ed d
ispl
acem
ent t
hat a
re a
ll
zero
sex
cept
for
a“1
”on
the
r’th
row
:.T
he
resu
lting
for
ce s
een
by
eac
h of
the
othe
r g
ener
aliz
ed d
egre
es o
f
free
dom
are
the
r’th
colu
mn
of.
Fr
q r∂∂
1 2---K
stq sq
tt∑
s∑
–
Krs
q ss∑–
==
q{} r
0.
1.
.0
T=
K
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
19
An
Exa
mpl
e P
rob
lem
of I
nvo
lvin
g th
eS
tiffn
ess
Mat
rix
Lets
con
side
r a
stru
ctur
esi
mila
rto
the
disk
-sha
ftsy
stem
inM
eiro
vitc
h.
The
stra
inen
ergy
inth
esh
aft
isco
mpu
ted
inte
rms
ofth
edi
ffer
ence
sin
the
rota
tions
at t
he d
isks
.
L1
L2
L4
L3
J 1,θ
1J 2
,θ2
J 3,θ
3
G, J
s
V1 2---
GJ s
θ 12
L1
-----
θ 2θ 1
–(
)2
L2
--------
--------
--------
θ 3θ 2
–(
)2
L3
--------
--------
--------
θ 32
L4
-----
++
+=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
20
An
Exa
mpl
e P
rob
lem
of I
nvo
lvin
g th
eS
tiffn
ess
Mat
rix
Our
stif
fnes
s m
atrix
is
Not
e th
at th
eth
ele
men
t of
is th
e to
rqu
e fe
lt on
dis
k d
ue to
a
unit
rota
tion
impo
sed
on d
isk
.
Kij
[]
θ iθ
j∂
2 ∂∂V
GJ s
1 L1
-----
1 L2
-----
+1 L2
-----
–0
1 L2
-----
–1 L2
-----
1 L3
-----
+1 L3
-----
–
01 L3
-----
–1 L3
-----
1 L4
-----
+
==
ij,
Ki
j
Advanced Vibrations
/home/djsegal/UNM/VibCour se/slides/11/30/98 Copyright Dan Segalman, 1998
21
Another Example In volving theStiffness Matrix
Consider acantile vered beam f orwhic h we postulate adisplacement fieldwith tw o freegeneraliz edcoor dinates:
The strain ener gy in the beam will be
,
and
EI, m
L
y x t,( ) A1 t( ) f 1 x( ) A2 t( ) f 2 x( )+=
V A1 A2,( ) EI2------ y ′ ′( )2
xd
0
L
∫=
12--- A1 t( )2
I4 2A1 t( )A2 t( )I5 A2 t( )2I6++[ ]=
I4 EI f 1′ ′ x( )( )2xd
0
L
∫= I6 EI f 2′ ′ x( )( )2xd
0
L
∫=
I5 m f 1′ ′ x( ) f 2′ ′ x( ) xd
0
L
∫=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
22
Ano
ther
Exa
mpl
e of
Mas
s M
atrix
Con
tinue
d
Our
stif
fnes
s m
atrix
isK
ij[
]A
iA
j∂
2 ∂∂V
EI
I 4I 5
I 5I 6
==
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
23
Mat
rix M
ultip
licat
ion
A s
hor
t re
vie
w
Say
and
are
col
umn
vect
ors
of le
ngth
, the
n a
re
defin
ed to
be
. Thi
s is
the
inne
r pr
oduc
t of a
lgeb
raic
vec
tor
s.
Say
is a
n b
y m
atrix
, the
n th
e m
atrix
vec
tor
prod
uct
is
defin
ed s
o th
at th
eth
ele
men
t of
is
Com
bini
ng th
e ab
ove
two
defin
ition
s:
ab
NbT
aaT
b=
a rbr
r1
=N ∑
AN
NA
a
rA
aA
rsa s
s1
=N ∑î
aTA
bA
rsa rb
ss
1=N ∑
r1
=N ∑
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
24
Mat
rix M
ultip
licat
ion
A s
hor
t re
vie
w
The
pro
duct
of t
wo
mat
rices
is d
efine
d in
a s
imila
r w
ay.
Say
is a
lso
an b
y m
atrix
als
o, th
en th
e pr
oduc
t is
defi
ned
as
The
tran
spos
e of
a m
atrix
is th
at w
hic
h is
obt
aine
d b
y re
vers
ing
the
orde
r of
the
indi
ces:
. T
he tr
ansp
ose
of p
rod
ucts
is
foun
dto
be.
In
mat
rix n
otio
n:
BN
NA
B
AB
() r
sA
rtB
tst
1=N ∑
=
AT
() r
sA(
) sr
=
AB
()T
() r
sA
B(
) sr
Ast
Btr
t1
=N ∑B
TA
T(
) rs
==
=
AB
()T
BT
AT
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
25
Fle
xibi
lity
Mat
rix(L
ets
star
t usi
ng m
atrix
not
atio
n)
In th
ose
case
s w
here
ther
e ar
e no
rig
id b
ody
mot
ions
, the
stif
fnes
sm
atrix
is n
onsi
ngul
ar, a
nd w
e ca
n de
fine
a fle
xibi
lity
mat
rix
.
In th
is c
ase
we
obse
rve
that
sin
ce,
.
The
str
ain
ener
gy m
ay
now
be
exp
ress
ed in
term
s of
for
ce
Som
etim
es v
ibra
tion
prob
lem
s ar
e fo
rmul
ated
in te
rms
of th
e fle
xibi
lity
mat
rix, t
houg
h th
e us
e of
dis
plac
emen
t-ba
sed
finite
ele
men
ts m
akes
flexi
bilit
y f
orm
ulat
ions
dec
reas
ingl
y po
pula
r.
Dis
cuss
ion
of th
e fle
xibi
lity
mat
rix is
pre
sent
ed h
ere
for
com
plet
enes
s.W
e sh
all d
o lit
tle m
ore
with
it in
this
cla
ss.
aK
1–= F
Kq
–=
qa
F–
=
V1 2---
qTK
q1 2---
aF
–()T
Ka
F–(
)1 2---
FT
aF
==
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
26
Mat
rix T
rans
form
atio
ns
Say
that
we
have
deriv
edth
efo
llow
ing
mat
rixre
pres
enta
tion
ofa
linea
r
syst
em:
whe
re th
e co
lum
n ve
ctor
con
tain
s al
lex
tern
ally
app
lied
load
.
No
w, s
ay w
e se
lect
ano
ther
set
of g
ener
aliz
ed c
oord
inat
es w
ith w
hic
h
we
can
spec
ify th
e fir
st s
et:
. Let
s re
-der
ive
our
gove
rnin
g
equa
tions
in te
rms
of.
Kin
etic
Ene
rgy
:
Str
ain
Ene
rgy
:
Pot
entia
l Ene
rgy
of L
oadi
ng:
Mq
Kq
+F
=F
qT
β=
β
T1 2---
qTM
q1 2---
Tβ
()T
MT
β(
)1 2---
βTT
TM
T(
)β=
==
V1 2---
qTK
q1 2---
Tβ
()T
KT
β(
)1 2---
βTT
TK
T(
)β=
==
Aq–
TF
Tβ
()T
F–
βTT
TF
()
–=
==
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
27
Mat
rix T
rans
form
atio
nLa
gran
ge
Equ
atio
n in
Ne
w S
yste
m
beco
mes
whe
re,
, and
Not
e th
at th
e m
ass
and
stiff
ness
mat
rices
rem
ain
sym
met
ric a
nd a
tle
ast p
ositi
ve s
emi-d
efini
te.
Ifis
ano
nsin
gula
r,s
quar
em
atrix
,th
etr
ansf
orm
atio
nis
calle
da
cong
ruen
ce tr
ansf
orm
atio
n of
A.
tdd
β r∂∂T
β r∂∂T–
β r∂∂V+
β r∂∂A –=
0
Mβ
Kβ
+F
=
MT
TM
T=
KT
TK
T=
FT
TF
=
TT
TA
T
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
28
Mat
rix T
rans
form
atio
n
We
ofte
n us
e su
ch
tran
sfor
mat
ions
whe
n in
trod
ucin
g co
nstr
aint
s to
asy
stem
. Her
e w
e co
nsid
er tw
oin
depe
nden
t pen
dulu
ms.
The
kin
etic
and
pote
ntia
l ene
rgi
es a
re
and
The
pot
entia
l ene
rgy
of l
oadi
ng is
The
line
ariz
ed e
quat
ions
of m
otio
n ar
e
R1
R2
m1
m2
F1
F2
Tm
1 2------
R12θ 12
m2 2------
R22θ 22
+=
Vm
gR
1θ 1
mg
R2
θ 2()
cos
–co
s–
= AF
1R
1θ 1
F2R
2θ 2
––
=
m1R
120
0m
2R
22
θ 1 θ 2
mg
R1
0
0m
gR
2
θ 1 θ 2+
F1R
1
F2R
2
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
29
Mat
rix T
rans
form
atio
nLe
ts a
dd
a co
nstr
aint
Con
side
r th
e ef
fec
t of c
onne
ctin
g th
etw
o m
asse
s w
ith a
mas
sles
s r
od. T
his
has
the
eff
ect o
f req
uirin
g
, per
mitt
ing
us to
writ
e
.
Defi
ne. T
hen
R1
R2
m1
m2
F1
F2
R1θ 1
R2θ 2
=
θ 1 θ 2
1
R1
R2
⁄θ 1
= T1
R1
R2
⁄=
M1
R1
R2
⁄m
1R
120
0m
2R
22
1
R1
R2
⁄m
1m
2+
()R
12=
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
30
Mat
rix T
rans
form
atio
nW
ith a
con
stra
int
The
stif
fnes
s m
atrix
is
and
the
appl
ied
for
ce is
K1
R1
R2
⁄m
1g
R1
0
0m
2g
R2
1
R1
R2
⁄=
m1g
R1
m2g
R12
R2
------
+=
F1
R1
R2
⁄F
1R
1
F2R
2
R1
F1
F2
+(
)=
=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
31
Mat
rix T
rans
form
atio
nW
ith a
con
stra
int
The
res
ultin
g eq
uatio
n f
or is
The
ad
ditio
n of
a c
onst
rain
t red
uced
the
num
ber
of a
ctiv
e de
gree
s of
free
dom
by
one
.
θ 1
m1
m2
+(
)R12
[]θ
1m
1g
R1
m2g
R12
R2
------
+
θ 1+
R1
F1
F2
+(
)=
Ad
van
ced
Vib
rati
on
s
/hom
e/dj
sega
l/UN
M/V
ibC
our
se/s
lides
/Lec
ture
10.fr
m11
/30/
98C
op
yrig
ht
Dan
Seg
alm
an, 1
998
32
Nex
t Tim
e
Sho
rt O
rient
atio
n
LAB
TO
UR