Part ElasticContinua
from shaft machine
Rotating
#I : :
-
Chapter 8 : Vibrations
of elastic continua
8 11 Vibrations
.
Longitudinal
↳ ocial
deformation :
g(x ,
El
↳ assumed consection I to
remains
plain ,
x-axis
has
body I corup.
a
and
eigenfree
↳ a DOF
mode shapes
2 ~
ext .
force per unit
The law newton PJX tdX
length
,
□
P(x
, El
ー
JAkd =
S + -Sp SVx)
π
A 세
* 5+
:d +
mass
□
acc.
+
PA (x )d
↳
Uning Hooks' law : S = Alo = AE ∞
→
PArA 1 = AA ) d +录 1 ← P 1x , t )
↳ assume Akx1 = A
=
PA = AE t
PS + ,ε 1
も 崎器
節
+
s
a
I =
propagation speed) Right traveling waves
>
-
if no load
p(x ,
) = 0-broutin is umb
of g
Left traveling waves
t) + 1
g(x g(x a
= +
,
=>
equation of motion is solved
for undamped ,
unloaded
body
natural form
of the body's
In me
↳
free vib .
modes
,
may
take :
t) Y(x) Jai lin we bic wt)
g (x
= +
,
↓
referred to as
eigen function
↳
represents mode shaps
of the
body corresp .
To wi
Subst gr t im
,
wave
eg
(
.
a[yx)(ainimwt
↳
T4Ma ; nimwt
+ bicatl) =
+ Gicawt))
-
ω㎡ 4( x ) =
@
2 *
1 +
器
4t 1 = 0
, = Y(x) =
Pein(x) +
& c( * x) - 3 unknowns.*
follow frm BCI
* example :
fer- fier shaft 9( ,
t) =
Y/X) (a ; nimwt + bi Cawt
- has no kinematic restrictions on BS such that no acid load S :
SA , t ) = AE =
AE SQisinwtt GiCswt )
S 10 E= SIe, )= mo oct .ocios
BC : I
force
)
0
, t
to keep it in
place I Mode y
_
ㅇ
앱 ] × (e ) = = 0
0 Ci 紫
= 0
= <i = 0
with
h htt =- Cider Com ex- diffline-
" hersel =o -
didline = ·
= sin 器 e = 0 =>
wid = i π
= wi
=
te ℃= §; i =
a.
= fundamental
…
elastic
eigenfrg. £
= Wi
not i
rigidmod
=
=
0
e
↳ ans
eigenfunctin T: (x) corup .
To
every Wi ?
Y, /) =
di Carlix) =
di
3
ca
(
just scoling <t
X
二
, &
example :
fixed-free
end
shaft 튿 e
x
ㄠ
B new read :
g10 ,
t = 0
=0 = 0 = di = 0 ㅇ
9 (x ,
t) =
Mala;limwt + b; cut)
Sleit 1 =① AE
19 = 0
)
Yor ) = Cisin { + 1 tdiCos (袋 +
β
더
내윙 = 0
← Ci 器 (≈ e 1 =
0
=→ ae = (2 i + u1 是
= Wi = 12 iti ) =
2i + t 花, i - 02 …
-
→ φ/A ) =
Cisin ( ( 2 i + 11 花 + )
-91, +) = limex : niMWit + : Co WE
&
example Complicated :
BI
at x 0 :
fixed 910 )
=
=> = 0
,
→ ψ10 | = 0 => di = 0
EA φ=µ
nnn
of = : SIe , GH-
eking
e A t 1 4IA ) ( Q : AIn oot tGi COWt )
9
t -
.
fra di Cos
M) =
Cinim x +
器 ( @☆ tt bi owt ]
| At
* φ (x , t) = Ain + sin ω
v [ ai un Go ω t ] ' Cas sinwt ott wa ε ]
*
器 bǒ )= nin to
ω
- t+ - +
& Et/sinwtcw) =
-Mein Freiwi
Es
EAC =
Mweinksin
AEW W =MW-klein horto be solved
numerically
to cale
eigenvals
E k
now
analyse following case : = o s no
spring
= ,
A Co = Mwanin 路 = AEE
a
=µ WI ton 路
→ =
n 路 ton 路
5
C ,
02 =
"
짬
e
=
waton
〜
에
es
n
m mae
_
e
心
心
from shaft machine
Rotating
#I : :
-
Chapter 8 : Vibrations
of elastic continua
8 11 Vibrations
.
Longitudinal
↳ ocial
deformation :
g(x ,
El
↳ assumed consection I to
remains
plain ,
x-axis
has
body I corup.
a
and
eigenfree
↳ a DOF
mode shapes
2 ~
ext .
force per unit
The law newton PJX tdX
length
,
□
P(x
, El
ー
JAkd =
S + -Sp SVx)
π
A 세
* 5+
:d +
mass
□
acc.
+
PA (x )d
↳
Uning Hooks' law : S = Alo = AE ∞
→
PArA 1 = AA ) d +录 1 ← P 1x , t )
↳ assume Akx1 = A
=
PA = AE t
PS + ,ε 1
も 崎器
節
+
s
a
I =
propagation speed) Right traveling waves
>
-
if no load
p(x ,
) = 0-broutin is umb
of g
Left traveling waves
t) + 1
g(x g(x a
= +
,
=>
equation of motion is solved
for undamped ,
unloaded
body
natural form
of the body's
In me
↳
free vib .
modes
,
may
take :
t) Y(x) Jai lin we bic wt)
g (x
= +
,
↓
referred to as
eigen function
↳
represents mode shaps
of the
body corresp .
To wi
Subst gr t im
,
wave
eg
(
.
a[yx)(ainimwt
↳
T4Ma ; nimwt
+ bicatl) =
+ Gicawt))
-
ω㎡ 4( x ) =
@
2 *
1 +
器
4t 1 = 0
, = Y(x) =
Pein(x) +
& c( * x) - 3 unknowns.*
follow frm BCI
* example :
fer- fier shaft 9( ,
t) =
Y/X) (a ; nimwt + bi Cawt
- has no kinematic restrictions on BS such that no acid load S :
SA , t ) = AE =
AE SQisinwtt GiCswt )
S 10 E= SIe, )= mo oct .ocios
BC : I
force
)
0
, t
to keep it in
place I Mode y
_
ㅇ
앱 ] × (e ) = = 0
0 Ci 紫
= 0
= <i = 0
with
h htt =- Cider Com ex- diffline-
" hersel =o -
didline = ·
= sin 器 e = 0 =>
wid = i π
= wi
=
te ℃= §; i =
a.
= fundamental
…
elastic
eigenfrg. £
= Wi
not i
rigidmod
=
=
0
e
↳ ans
eigenfunctin T: (x) corup .
To
every Wi ?
Y, /) =
di Carlix) =
di
3
ca
(
just scoling <t
X
二
, &
example :
fixed-free
end
shaft 튿 e
x
ㄠ
B new read :
g10 ,
t = 0
=0 = 0 = di = 0 ㅇ
9 (x ,
t) =
Mala;limwt + b; cut)
Sleit 1 =① AE
19 = 0
)
Yor ) = Cisin { + 1 tdiCos (袋 +
β
더
내윙 = 0
← Ci 器 (≈ e 1 =
0
=→ ae = (2 i + u1 是
= Wi = 12 iti ) =
2i + t 花, i - 02 …
-
→ φ/A ) =
Cisin ( ( 2 i + 11 花 + )
-91, +) = limex : niMWit + : Co WE
&
example Complicated :
BI
at x 0 :
fixed 910 )
=
=> = 0
,
→ ψ10 | = 0 => di = 0
EA φ=µ
nnn
of = : SIe , GH-
eking
e A t 1 4IA ) ( Q : AIn oot tGi COWt )
9
t -
.
fra di Cos
M) =
Cinim x +
器 ( @☆ tt bi owt ]
| At
* φ (x , t) = Ain + sin ω
v [ ai un Go ω t ] ' Cas sinwt ott wa ε ]
*
器 bǒ )= nin to
ω
- t+ - +
& Et/sinwtcw) =
-Mein Freiwi
Es
EAC =
Mweinksin
AEW W =MW-klein horto be solved
numerically
to cale
eigenvals
E k
now
analyse following case : = o s no
spring
= ,
A Co = Mwanin 路 = AEE
a
=µ WI ton 路
→ =
n 路 ton 路
5
C ,
02 =
"
짬
e
=
waton
〜
에
es
n
m mae
_
e
心
心
