Part I : MDOF : Theorie
Chapter 5 multi-degree-of-freedom
: models : modal
analysis
1) Free undamped system
5 vibrations
.
of an
m
Mij +
kg =
f(t) ge
mo Rm
I undamped of fl ε )m' ε 1
M =
0(SPD)
> ( =
mass motio)
K = k'To /SPSD) ( =
stiffues motioc (
=
first consider
f(t) =
o
free vibration
(1)
choosing Mg
(9
and =>
+
kq = 0
~ We
suggest a solution
of form q(t)
=
e; Sintit + p, ) (2)
↳
2; EIRM = vector
of time
independent amplitudes
This that whole
Assumes
system is
Capable of vibrating at a
single frequency Wi
~ Subst .
(2) im (1) = M)-e ;
w ! ein (wit + d, 1) + Ke ; einwit + di =
0
> (-wim + kle ; sintwit + di 1 =
0
X
non-trivia solution if det (-mwi + x) =
o
~ The vector 2;
follows from eq : [K-wiM]e ; =
↳
defines a quadratic
eigemede general:
[ K +λ QuJe =
0
volue X ∞
eigen depende
on motrices K and M
in the
M= M30 y--Xi
>
-
i 1,...
case =
Ijwi =
purely imaginary pair
~ We assume M-volver to be distinct and ordered as O <We wa ...
k'To
in case K = some
eigen
voluer
equal Zero .
The value the of the and
eigenfrequencies system
↳
Wi ,
i =
1 .
-
-
m or
modd vectore
2; 1 = 1 -. m ar
corresponding eigenvectore or
There
subjected to an
artitory scaling factor
,~ the
free vibration
of an
undamped multi-degree-of-freedom syst .
Consists
of solution
of form
9 1t )
: 1 续惜
.
/t ) 9 y 1t = Cinin( witt⑥ ) n( wit +: 1
g
=
)
:
9m /6 ) 2
; (m)
the the
↳
emplitude of vibration of every Coord gr is
Corresponding Camp .
C; /)
of eigenvector
two and
- ratio
of vibration of cord .
9/t) g, (t)
=>k
= implie that both word thea
reach
extreme ?
voluer
simultaneously their static
=>
they pare simultaneously trough equil .I
↳
If signe equal ar > in
phose
signs opposite
vibrate out of phase
g
/t) =
e
,
Sin (Wit +
d ; ) moder of vibrations
Cbei #mode Shape L EIRM
a
represents spatial position of the
generalized
↳ cid
W .
r .
E .
each other
if sys
to me
of it's vibration moder
- modes
of an
undemped system
- real moder
-fre response to an
arbitrary set
of initia condition
9 and -
or
will be linear
All modes
camb .
of rans
m
=>
glt)
=
[e ; sin (wit + dil
i =^
↳
ecoling depende an initial conditions
go and
, exampl two inverted
pendulums
A
m m 2
daf gr =
(1) /generised Coordinate
k
l M나
그들 . 8
-> Og T =
= m(201k2 tm(982)2 +
·
튿 X//
V
=
Ik Ekt +
+
Ekimo-Esino-mg(l-(s01)-mg/l-lad
if we consider small vibrations
→ V ≈ k5 θ☆+ kt θ☆+ 主K 0- {
0z )
mge θ
_
mge
く = T V -
↳repes
最 ( 最階
{( ork
。 時 r -時2 I # ngeor - me 2 oi= 0
}
‰→ - k= 8e-kIE01-20a)tEl +
mglos-mez8z = o
=> mecoI shit hearingslos - mencore>
We
-
> want
form M +
kg = 0 =
M =
(Ma
ttken -nge k
( 쯤 kE-mge
µ
崎
K =
- ke kf +
det (k-MWE)
↳
eigenfreq .
es solution of = o
Es k++ bE mgs- medr2
_
-
를
ke
를 -
ke ↳++ bE mgs- mer2
_
) R
→ ( kttke 是
-
mge - me
ω
2 β
_ k塔 =
0
=>
( kt mge -
-
mea ω
a
) / kt +2 k 路 mge _
-
me
2
ω
2
) = 0
n
2 mge
Et-me let +
=>
We =
on We =
mea
支
支
麦
支
Chapter 5 multi-degree-of-freedom
: models : modal
analysis
1) Free undamped system
5 vibrations
.
of an
m
Mij +
kg =
f(t) ge
mo Rm
I undamped of fl ε )m' ε 1
M =
0(SPD)
> ( =
mass motio)
K = k'To /SPSD) ( =
stiffues motioc (
=
first consider
f(t) =
o
free vibration
(1)
choosing Mg
(9
and =>
+
kq = 0
~ We
suggest a solution
of form q(t)
=
e; Sintit + p, ) (2)
↳
2; EIRM = vector
of time
independent amplitudes
This that whole
Assumes
system is
Capable of vibrating at a
single frequency Wi
~ Subst .
(2) im (1) = M)-e ;
w ! ein (wit + d, 1) + Ke ; einwit + di =
0
> (-wim + kle ; sintwit + di 1 =
0
X
non-trivia solution if det (-mwi + x) =
o
~ The vector 2;
follows from eq : [K-wiM]e ; =
↳
defines a quadratic
eigemede general:
[ K +λ QuJe =
0
volue X ∞
eigen depende
on motrices K and M
in the
M= M30 y--Xi
>
-
i 1,...
case =
Ijwi =
purely imaginary pair
~ We assume M-volver to be distinct and ordered as O <We wa ...
k'To
in case K = some
eigen
voluer
equal Zero .
The value the of the and
eigenfrequencies system
↳
Wi ,
i =
1 .
-
-
m or
modd vectore
2; 1 = 1 -. m ar
corresponding eigenvectore or
There
subjected to an
artitory scaling factor
,~ the
free vibration
of an
undamped multi-degree-of-freedom syst .
Consists
of solution
of form
9 1t )
: 1 续惜
.
/t ) 9 y 1t = Cinin( witt⑥ ) n( wit +: 1
g
=
)
:
9m /6 ) 2
; (m)
the the
↳
emplitude of vibration of every Coord gr is
Corresponding Camp .
C; /)
of eigenvector
two and
- ratio
of vibration of cord .
9/t) g, (t)
=>k
= implie that both word thea
reach
extreme ?
voluer
simultaneously their static
=>
they pare simultaneously trough equil .I
↳
If signe equal ar > in
phose
signs opposite
vibrate out of phase
g
/t) =
e
,
Sin (Wit +
d ; ) moder of vibrations
Cbei #mode Shape L EIRM
a
represents spatial position of the
generalized
↳ cid
W .
r .
E .
each other
if sys
to me
of it's vibration moder
- modes
of an
undemped system
- real moder
-fre response to an
arbitrary set
of initia condition
9 and -
or
will be linear
All modes
camb .
of rans
m
=>
glt)
=
[e ; sin (wit + dil
i =^
↳
ecoling depende an initial conditions
go and
, exampl two inverted
pendulums
A
m m 2
daf gr =
(1) /generised Coordinate
k
l M나
그들 . 8
-> Og T =
= m(201k2 tm(982)2 +
·
튿 X//
V
=
Ik Ekt +
+
Ekimo-Esino-mg(l-(s01)-mg/l-lad
if we consider small vibrations
→ V ≈ k5 θ☆+ kt θ☆+ 主K 0- {
0z )
mge θ
_
mge
く = T V -
↳repes
最 ( 最階
{( ork
。 時 r -時2 I # ngeor - me 2 oi= 0
}
‰→ - k= 8e-kIE01-20a)tEl +
mglos-mez8z = o
=> mecoI shit hearingslos - mencore>
We
-
> want
form M +
kg = 0 =
M =
(Ma
ttken -nge k
( 쯤 kE-mge
µ
崎
K =
- ke kf +
det (k-MWE)
↳
eigenfreq .
es solution of = o
Es k++ bE mgs- medr2
_
-
를
ke
를 -
ke ↳++ bE mgs- mer2
_
) R
→ ( kttke 是
-
mge - me
ω
2 β
_ k塔 =
0
=>
( kt mge -
-
mea ω
a
) / kt +2 k 路 mge _
-
me
2
ω
2
) = 0
n
2 mge
Et-me let +
=>
We =
on We =
mea
支
支
麦
支
