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7/24/2019 Kuliah Ke-3 FisKel-Metoda Energy
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FISIKA KELAUTANMO 14
mahmud mustain
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Kuliah ke-3
Metoda Energy
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Materi...
Dasar-dasar metoda energi gerakan enda!
Metoda Rayleigh, kerja virtual danLagrange,s"stem #inear dan non-#inear$respon bebas dan respon terpaksa, respondomain waktu
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Energy conservationKinetic energy with respect to inertial
reference
2
2
1
2
1
2
Constant
U W kx
T mx
T U
= =
=
+ =
&
x
!urdue "eroelasticity
https://engineering.purdue.edu/.../AAE556-Lectures_2..
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#ime rate of change ofenergy
x
( )
( )2 2
Constant
d0
d 1 1 2 02 2 2
T U
T Udt
mx kx x mx kxdt
+ =
+ =
+ = + =
& & &&
$!urdue "eroelasticity
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%agrangian mechanics for conservativesystems with a nite number of degrees of
freedom
'ene the %agrangianas %(#-)
*oordinates +, +, +n #hese are called
generali/ed coordinates
0
i i
d L L
dt x x
= &
2 21 1
2 2
22
2
2
0
i
i
L T U mx kx
L mxx
Lkx
xmx kx
= =
=
=
+ =
&
&&
&&
0!urdue "eroelasticity
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*ontinuous systems and the 1ayleigh-1it/method
- a beam e+ample *ompute kinetic energy and strain energy
22
2
2
1 1
2 2
wT Aw dy U EI dy
y
= = &
w(y,t)
2!urdue "eroelasticity
The internal forces areaccounted for in the strain
energy portion of L=T-U
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*onverting a continuous system to a system with anite number of degrees of freedom using assumed
deection shapes
( ) ( )
( ) ( )
( ) ( )
2
2
( , ) i i
i i
i i
w y t a t y
w
a t yy
wa t y
t
=
=
=
&
( ) ( ){ }
( ) ( ){ }
( ) ( ){ }
2
2
( , ) i i
i i
i i
w y t a t y
wa t y
yw
a t yt
= =
= &
4!urdue "eroelasticity
Approximate response as aseries of admissiblefunctions.
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*omputing the strainenergy in matri+ form( ) ( ){ }
( ) ( ){ }
( ) ( ){ }
2
2
( , ) i
i
i
w y t a t y
wa t y
y
w a t yt
= =
= &
( ) ( ){ } ( ) ( ){ }
( ) ( ){ } ( ) ( ){ }
2
2 " "
2
2
i i j j
i i j j
wa t y y a t
y
wa t y y a t
t
= =
& &
5!urdue "eroelasticity
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Kinetic energy and strainenergy
( ) ( ){ } ( ) ( ){ }
( ) ( ){ } ( ) ( ){ }
2 "
2
2
"21 1
2 2
1 1
2 2
i j
i
i j
i jj
wU EI dy a t a t
y
wT A
EI y y dy
A y y dydy a t a t t
= =
= =
& &
6!urdue "eroelasticity
% 7 i id if
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%agrange7s e8uations identifymass and sti9ness matri+
elements( ) ( ){ } ( ) ( ){ }
( ) ( ){ } ( ) ( ){ }
" "1
2
1
2
i i
i
i j
i j i
U a t a t
T
EI y y dy
A y ya t d a ty
=
=
& &
0i i
d L L
dt a a
= &
{ } { } { }0ij i ij im a k a + = &&
" "
ij i j ij i jm A dy k EI dy = = !urdue "eroelasticity
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"n e+ample - assumed deectionfunctions
:hape functions must at least satisfy
geometric boundary conditions
( ) 21 y y =
( )"1 2y =
" "ij i j ij i jm A dy k EI dy = =
!urdue "eroelasticity
y
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44
11 11
4
2
1
4
1 1
23 3 3
45
4 05
20 4
4 * * 0
.472 3.
5
518
Lm A y dy AL k EIL
M AL
MLEIL a
EI EI EIML ML M
MLa
L
EI L a
= = =
=
+ =
= =
=
+
&&
1esults ; %agrange7s e8uation
27% error
3!urdue "eroelasticity
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www.ecse.rpi.edu/.../Lecture%20.ppt
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%inear systems vs nonlinearsystems
%inearsystems
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%inear systems vs nonlinearsystems
%inearsystems
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%inear systems
%inear systems are systems that have a certainset of properties.
%inear systems are very nice ob=ects to studybecause of their regularity. >hy? >e needstructure.
:ystem
ic
outp
ut
input
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>hat is tricky about nonlinearsystems?
%"*K @A :#1)*#)1EB*annot take everything forgranted.
CE+istence and uni8ueness of solution to di9.
e8ns.
CAinite escape time
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>hy study nonlinearsystems?
%inearity is ideali/ation. E.g. a simple pendulum.
" lot of phenomena are only present in nonlinearsystems. Multiple countableF e8uilibria. >hy?
1obust oscillationsD where?
Iifurcations
*omple+ dynamics
>hy simulationis not always enough
>hy simulationis not always necessary
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