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GETARANGETARAN

MEKANISMEKANISoleh: SUKOCO

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MATERIMATERI

• SISTEM GETARAN

• SISTEM SATU DERAJAT KEBEBASAN

• GETARAN BEBAS TANPA PEREDAM• GETARAN PAKSA

• GETARAN BEBAS DUA DERAJAT

KEBEBASAN• GETARAN PAKSA DUA DERAJAT

KEBEBASAN

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SATUAN DASAR SISTEM SISATUAN DASAR SISTEM SI

PANJANG METER m

MASSA KILOGRAM Kg

WAKTU SEKON s

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GERAK OSILASIGERAK OSILASI

GERAK OSILASI: 

 ADALAH GERAK BOLAK BALIK

TERHADAP TITIK SETIMBANG

BEROSILASI PERIODIK= BERGETAR

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KLASIFIKASI GETARANKLASIFIKASI GETARAN

1. BERDASAR DERAJAT KEBEBASAN

» SATU DERAJAT KEBEBASAN

» GETARAN LEBIH DARI 1 DERAJATKEBEBASAN! DERAJAT KEBEBASAN

2. BERDASAR GAYA YG BEKERJA» GETARAN BEBAS

» GETARAN PAKSA

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KLASIFIKASI GETARANKLASIFIKASI GETARAN

3. BERDASAR REDAMAN

» GETARAN TAK TEREDAM

"TIDAK ADA REDAMAN#

» GETARAN TEREDAM "ADA REDAMAN#

4. BERDASAR PERSAMAAN GERAK

» GETARAN LINIER

"PERSAMAAN GERAK LINIER#» GETARAN TIDAK LINIER

"PERSAMAAN GERAK TIDAK LINIER# 

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2 DERAJAT KEBEBASAN2 DERAJAT KEBEBASAN

φ1

φ2

X1

X1 X2

K 1 K 2 K 3

φ

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1 Derajat Kebebasan 2 Derajat Kebebasan

3 Derajat Kebebasan 4 Derajat Kebebasan

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BERDASAR GAYA YG BEKERJABERDASAR GAYA YG BEKERJA

GETARAN BEBAS

  T$%&' &%& g&(& l)&* se+el&h g&(&

  &,&l %$le-&sGETARAN PAKSA

  A%& g&(& l)&* (&.g +e*)s me.e*)s

/e'e*0&

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GERAK ARMONIKGERAK ARMONIK

GERAK HARMONIK SEBAGAI

PRO2EKSI SUATU PARTIKEL 2ANG

BERGERAK MELINGKAR

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GERAK ARMONIKGERAK ARMONIK

2π

A

ωt

x

A Sin ωt

A

θ = ωt

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!er"#n$a%an s#&"an'an ( $#r)&)s*an

(1)Se$an'*an:

(2)

Ke+e"atan 'era* %ar&,n#* $#"er,-e% $en'an

$#.erens#as# "ersa&aan /10

t  A x   ω sin=

 f  T π ω 

π ω 

2

2

=

=

)2/(sin

cos

π ω ω 

ω ω 

+=

=

t  A x

t  A x

 

= xKecepatan Gerak Har!n"#

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!ERE!ATAN GERAK ARMONIK!ERE!ATAN GERAK ARMONIK

)sin(

sin

2

2

π ω ω 

ω ω 

+−=

−=

t  A x

t  A x

Harmonik Gerak Percepatan= x

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ONTO !ERSOALANONTO !ERSOALAN

15 S)&+) ge*&' h&*mo.$' mem-).(&$&m-l$+)%o 64!6 7m %&. -e*$o%e 6418 %e+$'5Te+&-'&. -e*7e-&+&. %&. 'e7e-&+&.

m&'s$m)m.(&9

!5 S)&+) &'sele*ome+e* me.).0)''&. s)&+)

/&.g).&. /e*ge+&* h&*mo.$' -&%& !H;4 %e.g&. -e*7e-&+&. m&'s$m)m 86 g5Te+&-'&. &m-l$+)%o.(&9 g=

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FOURIER /151360FOURIER /151360

TIAP GERAK PERIODIK DAPAT DIN2ATA KAN

OLEH DERET SINUS DAN 3OSINUS 2ANG

DIHUBUNGKAN SE3ARA HARMONIK

...sinsin.....coscos2

)( 221122110 +++++= t bt bt at aa

t  x   ω ω ω ω 

T dengan

  π  ω  

2: 1  =

1ω ω    nn   =

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Se-anj)tn7a8Se-anj)tn7a8

∫ 

∫ 

−

−

=

=

2

2

2

2

sin)(2

cos2

T 

T 

T 

T 

tdt t  xT 

b

tdt ω(x(t))T 

a

nn

nn

ω  

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29 GETARAN BEBAS29 GETARAN BEBAS

 A%&l&h Ge+&*&. (&.g +e*0&%$ +&.-&

*&.gs&.g&. %&*$ l)&* "'&*e.& s$s+em

mem-).(&$ m&ss& %&. el&s+$s$+&s#

' ">?@#

@mm

'' >

 >

WW

2.1 Pes!"!!n #e!$ %!n &e$'ensi n!t'! siste" e*!s "!ss!:

HUKUM GERAK NE$T%N (&'a)

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m

K >

W = mg

K > = W = mg

P&%& -os$s$ 'e%)& %$m&.& +e*0&%$ s$m-&.g&. @ :

m&'& g&(& (&.g /e'e*0& -&%& m&ss& "m# +s/

15 ' ">?@#

!5 W = mg

DITERA!KAN !ADA !ERUBAAN BENTUK!EGAS !ADA !OSISI KESETIMBANGAN

Catatan :

  Besaran Gaa! Ke"e#atan! Per"e#atan $e %a&a' ()an**a# #+s)t), 

UKUM NE:TON I

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Ses)a# $en'an )*)& Ne;t,n IISes)a# $en'an )*)& Ne;t,n II

m

k 

n

  =2

ω 

m

k n   =ω 

∑   =ma F  ∑   +∆−==   )(  xk W  F  xm &+&)

W k    =∆

kx xm   −=

0=+ kx xm 

%&.

De.g&. me.g&m/$l h)/).g&. *e')e.s$ s)%)+ .&+)*&l ".# %g ' %&. m :

0=+ kx xm 0=+   x x

mk 

02

=+   x x nω 

m&'&

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S,-)s# )&)&S,-)s# )&)&

A S"n *nt + B ,!# *nt 

K!n#tanta A -an B -"."t/n0 pa-a k!n-"#" Aa (!) -an !(!)

R//# 234

C!8

X = A Sin ωn t + B Cos ωnt

X(o) = A Sin 0 + B cos 0

X(o) = 0 + B . B = X(o)

+s$.B+7os A@....

  −=

6B1 A"6#@..

  −=

.

"6#@ A

=

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SEHINGGA :

+7osB@+s$.B@ ."6#.B"6#E

.

+=

PERI%DE NATURA5 %SI5ASI *nT 26

k 

mT    π  2=

DAN 'REKUENSI NATURA5

m

k 

T   f  n

π  2

11==

KARENA K7 0

∆=  g

π  21

n f   mmn f   ∆=  +,1-

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GETARAN TORSIONALGETARAN TORSIONAL

PIRINGAN DAPAT BERPUTAR BOLAK

BALIK

' = KEKAKUAN TERHADAP TORSI

J = MOMEN INERSIA POLAR MASSA

TERHADAP SUMBU 2ANG BERIMPITDENGAN BATANG

θ

'

J

Getaran T,rs#,na-

HUKUM NE$T%N II

PEN8E5ESAIANN8A :

SEHINGGA :

  j

k n  =ω  

'FFJ   −=

0$θθ  =+

t ω θ θ    cos0=

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T + U   K!n#tan

0)(   =+U T dt 

d 

METODE ENERGIMETODE ENERGI

D&l&m s)&+) s$s+em e.e*g$ +o+&l &%&l&h 'o.s+&.

E.e*g$ +e*%$*$ %&*$ e.e*g$ '$.e+$' "T # %&. e.e*g$ -o+e.s$&l

"U #

  T  = K&*e.& 'e7e-&+&..(&

  U  = E.e*g$ *eg&.g&. (&.g /e*)/&h /e.+)'

  = Ke*0& '&*e.& me%&. g&(& &+&) g*&$+&s$

E.e*g$ +o+&l 'o.s+&. m&'& -e*)/&h&. e.e*g$ &%&l&h .ol

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DARI KEKEKALAN ENERGI

 T1 + U1 = T2 + U2

!ADA GERAK ARMONIK

6

+

 TITIK SEIMBANG U1 = 6 T m&@ TITIK PUN3AK T! = 6 U m&@

SEHINGGA :

Ta9Ua9

3&+&+&.:

T = m !

U = ' @!

mg = ' >

K = mg >

1

2

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ONTO !ERSOALANONTO !ERSOALAN

2.1 S//SAKA SST/M P/#AS MASSA S/P/T

#AM4A #5 2.1

PEN8E5ESAIAN :

M

K >

T = KE = m @.! U = PE = K @!

T ? U = KONSTAN

.

21

21

2

2

12

2

1

2

212

21

0:

0.2..2.

.0)(

$onst!n

=+

=+

=+

=+

kx xm sehingga

 x xk  x xm

kx xmdt 

d 

kx xm

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2.2 T/T6KA 7/K6/S AT6A SST/M 8A#

9T66KKA S/P/T #AM4A 4/K6T

m

F

'

J

* 

R

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ONTO5ONTO KASUSONTO5ONTO KASUS

m

k 

kx xm

n  =

=+

ω 

0

l 

 I Gk 

 J 

k 

k  J 

 p

n

=

=

=+

ω 

θ θ    0

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..

..

l 

 g 

ml 

mgl 

mgl ml 

mgl  J 

n   ==

=+

=+

2

2

0

0

ω 

θ θ 

θ θ 

2

2

22

2

0)(

0)(

ml 

kamgl 

kamgl ml 

kamgl  J 

n

+=

=++

=++

ω 

θ θ 

θ θ 

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MASSA EFEKTIFMASSA EFEKTIF

 M!ss! Pe*!s = M M!$!, "!ss! 5!*i!n e*!s se!n;!n* dy 

s!"! %en*!n :

M&ss& Peg&s D$-e*h$+).g'&.

JIKA SIMPANGAN MASSA m = x, DAN SIMPANGAN DIANGGAP LINI!

MAKA "

D&. 'e7e-&+&..(&:

 M l dy

 x

l 

 y x

l 

 y

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GETARAN BEBAS TEREDAMGETARAN BEBAS TEREDAM

P&%& ge+&*&. /e/&s +e*e%&m4 -e*e%&m meles&- +e.&g&

s$s+em seh$.gg& m&'$. l&m& m&'$. /e*')*&.g

Be.+)' -e*s&m&&. ge*&' -&%& )m)m.(& &%&l&h

M@ ? % ? '@ = "+#..

(t) = !ayaPerangsang

d = !aya"edaman  G&(& Re%&m&. '&*e.& 'e'e.+&l&.

% = 3 @..

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SISTEM GETARAN BEBAS TERENDAMSISTEM GETARAN BEBAS TERENDAM

9!i %i!*!" 5en%! 5e5!s es!"!!n *e!$ 

6nt'$ *et!!n 5e5!s teen%!" 7(t) = 0

Se

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Sem)& h&*g& + %$-e.)h$ /$l&:

0ss m# mc$ =++ 999999999 / 35 0Pe*s&m&&. = Pe*s&m&&. K&*&'+e*$s+$'

( )m# $

$mc

$mc

%$s   −±−=999999999 / 35 0

SOLUSI

t  st  s !e Ae x   21 +=

999999999 / 35 0

A %!n 4 $onst!nt! %i

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• Penee#a"an per#aaan e;". en0/nt/n0kan-en0an Redaman Kritis Cc 

• B"a -"#kr""nan Per#aaan akare-aanna -"#e;/t Redaman Kritis

( )

( )   $nm# $

$mCc

m# $

$mC

ω

0

==

=−

n$mω#m$Cc   == 999999999 / 35< 0

N"a" #/at/ re-aan -"natakan -aa re-aankr"t"# #e."n00a

"" =ζ  999999999 / 3516 0

Ras)+ Re(aan

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Se')n**a

n$m

Cc

$m

C

&ω&   ==Seh$.gg& -e*s&m&&. %$.(&+&'&. %&l&m

(   ) n$%$ ω%&&s .−±−= 999999999 / 3511 0Ke+$g& 'e&%&&. *e%&m&. +e*g&.+).g

1 1 =1

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Pe.$.0&)&.

15 1464 Ke&%&&. ')*&.g +e*e.%&m Re%&m&. *$.g&. +e*0&%$

ge*&' Be*os$l&s$9en*!n "ens'5sit'si$!n Pes ( 3-11 ) $e ( 3-? ), Se

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#et!!n 4eosi!si @ 1,0

!5 1 6 4 Ke&%&&. /&.(&' +e*.%&m *e%&m&. /e*&+5Ge*&' +&' /e*os$l&s$4 D 64 %&. -e.(eles&$&.

)m)mtω%)&&(tω%)&&( n

$n

$

B'A'x  −−−−+− +=

%&$ω

x(ω%)&(&(x

%&$ω

x(ω%)&(&(x-

$

n

n

$

$

n

n$

−

−−−−=

−−++=

))

))%e.g&.

%&.

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/as)0 : ta$ a(a *etaran #er)+()$! ta$ a(a *etaran!

*era$na a#er)+()$ t)(a$ $e%a0) $e t)t)$

set)%an*

G/5 Ge*&' A-e*$o%$' 146

tω%)&&( n$

A'  −+−

tω%)&&( n$

B'  −−−

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5 = 146 Re%&m&. K*$+$s Ge*&' Te*e%&m K*$+$s

1 = ! = .

tωtω nn c'B)'(Ax   −− =+=Kon%isi !>! X(B) %!n X(B) %en*!n @=1

x(o).x(o)/tω(o)01x'x notω

n++=

  −

Gb9 354 Gera* Teren$a& Kr#t#s =>1?6

o

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!ENGURANGAN LOGARITMA!ENGURANGAN LOGARITMA

3&*& me.e.+)'&. 0)ml&h *e%&m&.%&l&m s)&+) s$s+em

&%&l&h me.g)')*l&0) -el)*)h&. os$l&s$/e/&s4 m&'$. /es&**e%&m&..(& m&'&

m&'$. /es&* -)l&l&0) -el)*)h&..(. C-1 !;' Pen*'!n*!n Bsi!si %en*!n en*'!n*!n o*!it"!

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Getaran Teren-a an0 D"natakan !e. per#?

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B$l& Pe*$o%e Re%&m&.

$

n   &%ω

$*

−=

B$l& 'e7$l 4   %&%   $ =−

M&'&$&%

&3

−=

2????????? ( @ 3 < )

Seh$.gg& &3   π 2= ????????? ( @ 3 @ )

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@9 REDAMAN OULOMB@9 REDAMAN OULOMB

T$m/)l m$s&l.(& -&%& gese'&. &.+&*& ! /$%&.g 'e*$.g

Bes&* gese'&. +$%&' +e*-e.g&*)h 'e7e-&+&. *el&+$4

mel&$.'&. +e*g&.+).g -&%& g&(& .o*m&l &.+&*& 'e%)&

/$%&.g

4N5* = = Koe$s$e. Gese'&. K$.e+$'

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# 

$5*)x(x

5*)x(x

)x(x# 

0)x5*(x%)#(x#x

%%

%%

$

%

$

%$%

%%

$

$%$

%$%

=−

=+−

=+−−−

−

−−

−

G% 2 Getaran Be%as (en*an Re(aan C+0+%

31  Ap"t/-! #etea. #eten0a. #"k/#

D0 p!#"#" e9tr" kecepatan n! ap"t/-! 91per/;a.an Ener0" k"net"k > -an kera an0 -"ak/kan pa-a >

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B"a Pr!#e-/r "n" -"/an0 /nt/k #eten0a. #"k/# ;er"k/tnaMaka -"per!e. pen0/ran0an Ap"t/-! a0" 2'-CkSe."n00a per/;a.an ap"t/-! per #"k/# :

# 

65*xx $%   =− 999999999 / @ 5 2 0

Gerak akan ;er.ent" ;"a ap"t/-! e;". kec" -ar" 7

G/5 8! me.).0)''&. /&h,& ge+&*&. /e/&s %e.g&.

*e%&m&. 7o)lom/4 &m-l$+)%o mel)*)h se7&*& l$.$e*

+e*h&%&- ,&'+) "g&*$s l)*)s#

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9 GETARAN !AKSA9 GETARAN !AKSA

S$s+em Ge+&*&. s&+) %e*&0&+ 'e/e/&s&. (g

me.%&-&+ g&.gg)&. %&*$ l)&* (g /e'e*0& -&%&

m&ss&5

G&(& +e*se/)+ +$m/)l &'$/&+: m&ss& +&' /&l&.7e4

'e+$%&'se$m/&.g&. -&%& mes$.4

s$m-&.g&. (&.g /e'e*0& -&%& l&.%&s&.4 &+&) g&(& -&'s& (&.g h&*mo.$' (&.g /e*s$&+

-e*$o%$' %&. %$'e+&h)$ *e')e.s$.(&

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GETARAN ARMONIK !AKSAGETARAN ARMONIK !AKSA

S)&+) s$s+em m&ss&4 -eg&s %&. -&*e%&mse-e*+$ g/5 14 %$'e.&'&. se/)&h g&(& l)&* o

s$. + se7&*& +e*)s me.e*)s

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Unt/k en0eta./" Ap"t/-! #"pan0an 9 -ankarakter"#t"k ;e#ar 9 ter.a-ap * -ar" -"a0ra;en-a ;e;a# per#aaan 0erak:

999999999 / 5 1 0

S!/#" k./#/# -en0an a#/#":

• kea-aan t/nak• rek/en#" * an0 #aa -en0an rek/en#"

ek#"ta#"

999999999 / 5 2 0

 #  = ap"t/-! !#"a#"

  = ;e-a a#a #"pan0an ter.a-ap 0aa ek#"ta#"

ωt  Fokxxmx ooo sin=++

)sin(   φ ω   −=t  #  x

φ 

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Den0an en#/;#"t/#" per#aaan 32 ke -aaper#aaan 31

 

 

 

 

 −

=

+−=

−2

1

22 )()(

ω 

ω φ 

ω ω 

mk 

tg 

mk 

 Fo

 #  999999999 / 5 3 0

999999999 / 5 4 0

Per#aaan 3< -an 3@ -apat -"natakanDaa ;ent/k tak ;er-"en#"

( )   ( )

  

 

 

 

 

−

=

+−=

−

k 

m

k 

k 

k m

k  Fo

 # 

2

2

1

t!n

1

1

22

ω 

ω 

ω ω 

φ 

999999999 / @ 0

999999999 / 0

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PERSAMAAN DI ATAS DAPAT DIN8ATAKAN DA5AMBESARAN

= *e')e.s$ g&(& -e.gg&.gg)

= *e')e.s$ -*$/&%$ s$s+em

= *e%&m&. '*$+$s =

= -e*/&.%$.g&. *e%&m&.

  = h&*g& *e%&m&. (&.g %$-&s&.g

ω 

n m   ω 2= km2

=ζ 

m

k n  =ω 

n

k 

k 

ω 

ω ζ 

ω ω 2=⋅=

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2

222

1

2

t!n

21

1

    

  −

   

  

 

=

 

 

 

 

 

 +

 

 

 

 

 

 −

=

n

n

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8/17/2019 POWER GETMEK 8-8-11

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