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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)


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

= =

= &


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

= =

& &



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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

= =

= =

& &


% 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



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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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